Fuzzy Approximate Reasoning toward Multi-Objective

Fuzzy Approximate Reasoning toward Multi-Objective Optimization Policy: Deployment for Supply Chain Programming M.H. Fazel Zarandi, Mosahar Tarimoradi, M.H. Alavidoost, Behnoush Shakeri Computational Intelligent Systems Laboratory Department of Industrial Engineering and Management Systems Amirkabir University of Technology Tehran, Iran

[email protected], [email protected], [email protected], [email protected] Abstract— To make a policy and decision for an appropriate set of optimizer algorithms is an important and controversial issue. It is significant especially when we want to consider more than a single objective and have to use multi-objective applications. The aim of this paper is to consider procedural fuzzy approximate reasoning to infer which one of the Multi-Objective Evolutionary Algorithms (MOEAs) could play a role in the suitable set as prevalent tool. The proposed procedure is put into practice for an invented biobjective programming in the supply chain and three numbers of similar applications from the same family, i.e. NSGA-II, NRGA, and PESA-II are deployed. Keywords— fuzzy approximate reasoning; optimization policy; supply chain; NSGA-II; NRGA; PESA-II.

I. INTRODUCTION The MOEAs are useful for big and complicated problems with more than a single objective in which one cannot find optimal solutions by the exact method. While being encountered with more than a single objective the selection for suitable MOEAs is vague, and also since the meta-heuristics act randomly, it is better to use fuzzy approximate reasoning ([1], [2], [3]and etc. ) which is used by practitioners in many areas ([4], [5], [4], and etc.). The issue is considered in the proposed procedure in this paper so that the functionality measurements by a couple of linguistic variables [6], as Quality and Time are associated as input variables for the reasoning process ([7], [8], [9], [10], and etc.). The contributions in this paper are organized as though, first the proposed procedural fuzzy approximate reasoning is explained. A bi-objective problem in supply chain programming is invented, and three MOEAs from the same family are used. The NSGA-II is introduced by Deb [11] as one of most used and propounded GA-based algorithms for solving multiobjective problems [12], the NRGA is presented by Jadaan using transformation of the NSGA-II selection strategy from the Tournament selection to the Roulette Wheel selection[13], and the PESA-II is

presented by Corne et al. to make NSGA-II faster and to mitigate its complexity [14]. II. PROCEDURAL APPROXIMATE REASONING As shown in Fig. 1, the approximate reasoning procedure in this paper consists of the actions to expand the meta-rule, and the inference based on the emerged fuzzy rule base. The premises and consequences of the rules need the documented experience of the MOEAs’ performance. Their performance approximation relying on indexes could be caught and also the membership functions parameters to have a fuzzy rule base by fuzzy arithmetic could be calculated. After these, the desires about the quality of results and the processing time determine which rules should be fired and what is an appropriate set of the MOEAs, and develop a control policy [15]

Fig. 1 Policy making based on fuzzy approximate reasoning

The fuzzy rule base is supposed that should be expanded using the Meta-Rule accordingly as follows: Meta-Rule: ̃ )  Suitable Subset of {B,C,D} From: Type(Time, QPF

Where: A: {NSGA-II, NRGA, PESA-II} B: Subset of A that has excellent value for Time ̃ C: Subset of A that has excellent value for 𝑄𝑃𝐹 ̃ D: Subset of A that has good value for Time and also 𝑄𝑃𝐹 Time, is linguistically valued as: Free, or Bounded, or Exigent ̃ , is linguistically valued as: Average, or High, or Excellent 𝐐𝐏𝐅

̃ ) as a fuzzy The Quality of Pareto Front (𝑄𝑃𝐹 operator which aggregates the values of multiple indexes, is one of the input variables in this meta-rule. ̃ for each one of the evolutionary The values of 𝑄𝑃𝐹 applications tend to the membership function parameters. Thus, the Triangular Fuzzy Number (TFN) and some criteria should be referred to. A TFN, as is graphically shown in Fig. 2, can be characterized by three parameters𝐴̃ = (𝐴1 , 𝐴2 , 𝐴3 ). TFN is used in this paper because of its computational simplicity in comparison with the other fuzzy numbers, as it is considered by Kaufmann [16], and more importantly that it matches with the semantics of the issue. The emerging results by each evolutionary application during the tuning and their value in each one of the indexes could be defined by a TFN. In other words, a TFN could be devoted to each index in each one of the MOEAs. To do so, the minimum value for each index is caught as𝐴1 , mean value for each index as𝐴2 , and finally, the maximum amount of each one as𝐴3 .

1.0

𝑨𝟏

𝑨𝟑

𝑨𝟐

nodes which are related to the customers to the right. The developed SC is axiomatized as follows: 1. It has an integrated structure consisting both of potential supplier and potential DCs to procure retailer demands for multitude commodities. 2. It has predefined numbers of suppliers and DCs with identified capacities. 3. The number of its retailers and their demands distribution are identified. 4. It operates in an uncertain circumstance, i.e. its main interior parameters as demands, lead-time, procure and transportation costs, and also holding costs of inventory for commodities are all supposed to be uniform random variables with identified average and variance (Table I). 5. Its DCs and suppliers are all supposed to be potentially operational at the beginning of the constructing network. 6. Its suppliers and DCs do their procuring, shipment, and holding duties perfectly. 7. Any retailer receives its demand for a specific merchandize only from one of the DCs. 8. Shortage cannot occur at the retailer nodes in any form. 9. More than one supplier can replete the demand of a specific DC. 10. More than one DC can replete the demand of each retailer. In other words, it is a Balanced Supply Chain Network (BSCN) in which the total capacity of all suppliers is greater or equal to the total consumption of all commodities, and the total procurement capacity of all commodities are equal to the total consumption of all end customers, and the total consumption of all commodities is equal to the total procurement of all commodities by suppliers.

Fig. 2 Schematic of a triangular fuzzy number

̃ helps to aggregate the emerged results The 𝑄𝑃𝐹 during the calib-ration relying on the quality/precision indexes. Based on the fuzzy arithmetic relations, as ̃ which is based on the mentioned formerly, the 𝑄𝑃𝐹 Yager Ordered Weighting Averaging (OWA) [17], could be calculated by (1) : 𝐿

̃ = ∑ 𝑤𝑖 . 𝑐𝑖 𝑄𝑃𝐹

(1)

𝑖=1

Where 𝑐𝑖 is ith index, and 𝑤𝑖 donates the weight or in better words, the priority of the ith index. III. CASE PROBLEM As a test problem, a supply chain problem is modeled. A couple of objectives are associated with the invented model. The first considers the total cost of SC and makes it minimum, whilst the second objective function maximizes the customer service level. As shown in Fig. 3, the minded SCN here is a three echelon one, consisting of suppliers to the left, distribution centers (DCs) in the middle, and retailer

Fig. 3 The considered network structure

The used notations in this model are listed in Table I, and the decision variables are also defined as shown in Table II. Based on the stock theory, the jth warehouse daily demand distribution for the kth commodity follows 𝑁(𝐷𝑗𝑘 , 𝜃𝑗𝑘 ). Where 𝐷𝑗𝑘 delegates the average daily demand of the jth warehouse for the kth commodity and 𝜃𝑗𝑘 is the daily demand variance of the jth warehouse for the kth commodity. The formulas for

calculation of 𝐷𝑗𝑘 and 𝜃𝑗𝑘 are as (2) while ∀ 𝑗 = 𝐼

𝐷𝑗𝑘

=

Subject to:

𝐽

∑ 𝑥𝑗𝑖𝑘 ≤ 1,

𝐼

∑ 𝜇𝑖𝑘 . 𝑥𝑗𝑖𝑘 , 𝑖=1

𝜃𝑗𝑘

=

∑ 𝜐𝑖𝑘 . 𝑥𝑗𝑖𝑘 𝑖=1

(2).

𝑥ijk ≤ uj ,

The expected value of the kth commodity lead-time delivery in the jth warehouse could be calculated by (3) while ∀ 𝑗 = 1,2, . . , 𝐽 , 𝑘 = 1,2, … , 𝐾:

∀ 𝑖 ∈ 𝑆𝐼 , 𝑗 ∈ 𝑆𝐽 , 𝑘 ∈ 𝑆𝐾 𝑘 ∑ 𝑦𝑚𝑗 ≤ 1,

(3).

∀ 𝑗 ∈ 𝑆𝐽 , 𝑘 ∈ 𝑆𝐾 𝑘 𝑦𝑚𝑗 ≤ 𝑧𝑚 ,

The average and variance of the specific kth commodity demand in lead-time for the jth warehouse are as (4)0 and (5), while ∀ 𝑗 = 1,2, . . , 𝐽 , 𝑘 = 1,2, … , 𝐾:

∀ 𝑚 ∈ 𝑆𝑀 , 𝑗 ∈ 𝑆𝐽 , 𝑘 ∈ 𝑆𝐾

∑ 𝑢𝑗 ≤ 𝑁

(4).

𝑗=1 𝑀

(5)

𝑚=1

𝑖=1 𝐼 𝐼

𝑖=1

Then𝑆𝑆𝑗𝑘 , the buffer quantity of the kth commodity for the jth warehouse could be calculated by (6) while

𝑖=1

𝑄𝑗∗𝑘 = √

+

.(7)

2. 𝐴𝑗𝑘 . 𝛽 ∑𝐼𝑖=1 𝜇𝑖𝑘 . 𝑥𝑗𝑖𝑘 ℎ𝑗𝑘

(8)

Where the used indices: i : Number of Retailers j : Number of Warehouses (DCs) k : Number of Commodities m : Number of Suppliers The mathematical model is expressed as follows: 𝐎𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝟏: 𝑓1 𝐽

𝑀

(9)

= Min ∑ 𝑔𝑚 . 𝑧𝑚 + ∑ 𝐹𝑗 . 𝑢𝑗 𝐾

{

𝑚=1 𝑀

𝑗=1 𝐽

𝐼

𝑘 𝑘 + 𝛽 ∑ ∑ ∑ ∑ 𝜇𝑖𝑘 . 𝑟𝑐𝑚𝑗 . 𝑥𝑗𝑖𝑘 . 𝑦𝑚𝑗 𝑘=1 𝑚=1 𝑗=1 𝑖=1 𝐽 𝐾 𝐼

+ 𝛽 ∑ ∑ ∑ 𝜇𝑖𝑘 . 𝑡𝑐𝑗𝑖𝑘 . 𝑥𝑗𝑖𝑘 𝑘=1 𝑗=1 𝑖=1 𝐾

𝐽

𝐼

𝐽

TABLE I. NOTATION USED IN FORMULATION

𝐼

𝑘 + ∑ ∑ ℎ𝑗𝑘 . 𝑧1−𝛼 . √ ∑ ∑ 𝐿𝑘𝑚𝑗 . 𝜐𝑖𝑘 . 𝑥𝑗𝑖𝑘 . 𝑦𝑚𝑗 𝑘=1 𝑗=1

𝑚=1 𝑖=1

𝐎𝐛𝐣𝐞𝐜𝐭𝐢𝐯𝐞 𝟐: 𝑓2 𝐽 𝐼 𝑘 𝑘 𝑘 𝑀 ∑𝐾 𝑘=1 ∑𝑚=1 ∑𝑗=1 ∑𝑖=1 𝜇𝑖 . 𝑥𝑗𝑖 . 𝑦𝑚𝑗 = Max { } 𝐾 𝐼 𝑘 ∑𝑘=1 ∑𝑖=1 𝜇𝑖

(19)

The objective function 1 (9) minimizes the total cost of setting up and operating the network, and the objective function 2 (10) maximizes the replenishing rate or service level. The constraint in (11) states that the ith retailer receives the kth commodity that could be satisfied just from one warehouse. The constraint in (12) specifies that the variables are bounded. The constraint in (13) enforces the kth commodity demand of the jth warehouse that could be procured just by one supplier. The constraint in (14) states that if the mth supplier is open, the jth warehouse will receive its demand from the mth supplier. The constraint in (15) indicates the maximum number of warehouses. The constraint in (16) specifies the maximum number of suppliers. The constraint in (17) ensures that the jth warehouse capacity is greater than the ith retailer demand and its buffer. The constraint in (18) enforces that the supplier capacity must be greater than the warehouse capacity. The constraint in (19) indicates that the variables are binary. Table I and Table II depict the used notations.

𝑖=1 𝑀

(18)

𝑘 𝑥𝑗𝑖𝑘 ∈ [0 , 1] , 𝑦𝑚𝑗 ∈ [0 , 1] , 𝑢𝑗 ∈ [0 , 1], 𝑧𝑚 ∈ [0 , 1]

Notation

𝐾

(17)

]

∀ 𝑚 ∈ 𝑆𝑀 , 𝑘 ∈ 𝑆𝐾

+ ∑ ∑ √2. 𝐴𝑗𝑘 . ℎ𝑗𝑘 . [∑ 𝜇𝑖𝑘 . 𝑥𝑗𝑖𝑘 ] 𝑘=1 𝑗=1

𝐼

}

(10)

μki υki

Meaning

=

𝑚=1 𝑖=1

∀ 𝑗 ∈ 𝑆𝐽 , 𝑘 ∈ 𝑆𝐾

𝑗=1 𝑖=1

The order point and optimum quantity of the j warehouse (𝑄𝑗∗𝑘 ) are as (7) and (8) while ∀ 𝑗 = 𝑆𝑆𝑗𝑘

(16)

𝐼

k ∑ [∑ 𝜇𝑖𝑘 . 𝑥𝑗𝑖𝑘 ] . 𝑦mj ≤ 𝑠m . zm ,

(6). th

𝐷𝑗′𝑘

[

𝐽

𝑆𝑆jk = 𝑧1−𝛼 . [√𝜃𝑗′𝑘 ]

𝑟𝑗𝑘

𝑀

𝑘 ∑ 𝜇𝑖𝑘 . 𝑥𝑗𝑖𝑘 + 𝑧1−𝛼 . √ ∑ ∑ 𝐿𝑘𝑚𝑗 . 𝜐𝑖𝑘 . 𝑥𝑗𝑖𝑘 . 𝑦𝑚𝑗 ≤ 𝑤𝑗 . 𝑢𝑗 ,

, 𝑘 = 1,2, … , 𝐾:

, 𝑘 = 1,2, … , 𝐾:

(15)

∑ 𝑧𝑚 ≤ 𝑅

𝜃𝑗′𝑘 = 𝐸𝑗𝑘 . 𝜃𝑗𝑘 = 𝐸𝑗𝑘 . ∑ 𝜐𝑖𝑘 . 𝑥𝑗𝑖𝑘

1,2, . . , 𝐽

(14)

𝐽

𝐼

∀ 𝑗 = 1,2, . . , 𝐽

(13)

𝑚=1

𝑚=1

𝐷𝑗′𝑘 = 𝐸jk . 𝐷𝑗𝑘 = 𝐸jk . ∑ 𝜇𝑖𝑘 . 𝑥𝑗𝑖𝑘

(12)

𝑀

𝑀

𝑘 𝐸𝑗𝑘 = ∑ 𝐿𝑘𝑚𝑗 . 𝑦𝑚𝑗

(11)

𝑗=1

∀ 𝑖 ∈ 𝑆𝐼 , 𝑘 ∈ 𝑆𝐾

Average Daily Demand from ith Retailer for kth Commodity Variance Daily Demand from ith Retailer for kth Commodity

Dimension

, 𝑘 = 1,2, … , 𝐾:

Distribution / Value

1,2, . . , 𝐽

U[70-120]

unit

U[10-25]

unit

hkj AkJ wj tcjik gm k rcmj

sm lkmj R N β

jth Warehouse Opening Fixed Cost jth Warehouse Holding Cost for kth Commodity jth Warehouse ordering cost for kth Commodity Potential Capacity of jth Warehouse Unit Cost of kth Commodity shipping from jth Warehouse to ith Retailer mth Supplier Fixed Cost to be Selected/Accept to Procure Cost of Procuring, Stocking and Shipping kth Commodity from mth Supplier to jth Warehouse Potential Capacity of mth Supplier Lead-time for kth Commodity from mth Supplier to jth Warehouse The Maximum Possible Number of Supplier The Maximum Possible Number of Warehouses The Number of Working-day Per Year

650

$

U[70-90]

$

5$

$

750

unit

U[10-15]

$

1500

$

U[65-80]

$

500

Monthly

U[2-3]

Day

50

unit

25

unit

220

Day

TABLE II. DECISION VARIABLES USED IN FORMULATION Variables 𝑥𝑗𝑖𝑘 ∈ [0 , 1] 𝑘 𝑦𝑚𝑗 ∈ [0 , 1]

𝑢𝑗 ∈ [0 , 1] 𝑧𝑚 ∈ [0 , 1] 𝑄𝑗𝑘 ≥ 0 𝑆𝑆jk ≥ 0 𝑟𝑗𝑘 ≥ 0

Definition th

1, If the Demand of k Commodity for ith Retailer is Satisfied by jth Warehouse, Else 0 1, If the Stock of kth Commodity for jth Warehouse is Procured by mth Supplier, Else 0. 1, If jth Warehouse is Open/Active, Else 0. 1, If mth Supplier is Selected for/Accept Procuring, Else 0. Optimum Quantity of kth Commodity for jth Warehouse Buffer Quantity of kth Commodity for jth Warehouse kth Commodity Order Point for jth Warehouse

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

CPT

300

350

400

450

500

Fig. 4 Elapsed Time measurement by CPT

As collation on the applications’ run time which is exposed in Fig. 4 shows, the NSGA-II and NRGA CPU Time are almost the same, whilst PESA-II needs less processing time rather than the others (almost two-thirds in comparison with the others). ̃ is a fuzzy It was formerly indicated that 𝑄𝑃𝐹 operator to aggregate the fuzzy measurement results. The expansion of the operator is presented as (20): ̃ = w1 × RNI ̃ + w2 × UD ̃ + w3 × Di ̃ + w4 𝑄𝑃𝐹 ̃ × QM (20) Where 𝑊𝑖 are determined as 𝑤1 = 0.2, 𝑤2 = 0.2, 𝑤3 = 0.2, 𝑤4 = 0.4 (by the authors). 1

QFP

NSGA-II NRGA PESA-II

0.8 0.6

Quality Metric (QM)

𝑗

𝑫𝒊 = ∑ max|𝑓𝑘𝑖 − 𝑓𝑘 | ; 𝑘=1

𝑖,𝑗

∀ 𝒊, 𝒋 = 1,2, … , 𝑛 |𝑃𝑇𝑙 ∈ 𝑃𝑇 ∗ | 𝑸𝑴𝒍 = |𝑃𝑇𝑙 | Where 𝑷𝑻∗ = Global Non-Dominated Points

Measures Pareto Fronts’ Diversity

Quality of Results [21]

0.4

0.9

1.4

1.9

̃ aggregation results Fig. 5 𝑄𝑃𝐹

̃ . Thus, the Fig. 5 shows the final value for𝑄𝑃𝐹 applications could be ranked visually as though NRGA is first, and it is followed by NSGA-II, and the last one would be PESA-II. Based on these computations, the parameters for input variables could be presented as Table IV, and up to now the premises of the rules are cleared. Now the meta-rule expansion needs approximation on the MOEAs performance during their tuning process as presented in Fig. 6 to Fig. 10. Thus, the consequences for rule-base could be extracted. ̃ MF PARAMETERS TABLE IV THE TIME AND 𝐐𝐏𝐅 NSGA-II

𝑛𝑂𝑏𝑗

Diversity (Di)

0

PESA-II

Uniformly Distribution (UD)

TABLE III PRIMED INDEX TOOLKIT Equation Description Needed Time for Elapsed Time Processing 𝑛 Ratio of Non𝑹𝑵𝑰 = Dominated Member 𝑛𝑃𝑜𝑝 Numbers to the Total While 𝒏𝑷𝒐𝒑 = Population [19] Number of Population 1 𝑼𝑫 = Measures Pareto 1 + 𝑆𝑆 Where 𝑺𝑺 = Fronts’ Uniformly Distribution [20] 2 1 √ ∑𝑛 (𝑑̅ − 𝑑𝑖 ) 𝑛−1 𝑖=1

0.2

Points

Index CPU Time (CPT) Ratio of Non-dominated Individuals (RNI)

0.4

Input Variable

IV. THE EXPERIMENTS To carry the proposed procedure of fuzzy approximate reasoning for the test problem into practice, the authors deployed the three similar applications from the same family, NSGA-II, NRGA, and PESA-II. Note that the applications parameters are tuned as proposed by Alavidoost [18]. The performance approximation and fuzzy evaluation of MOEAs need some indexes that are used for fuzzy measurement as presented in Table III.

550

NRGA

Fj

̃ QPF

LS C RS

0.483795667 0.554328183 0.902072635

0.44986916 0.69826150 1.67644410

0.56192220 0.852 1.41604649

Time

LS C RS

310.1630507 350.4072951 378.5182278

502.376267 510.498053 521.206007

497.930671 505.043966 522.094391

Emerged

Fig. 6 Line plot and boxplot for CPU Time

but less than those emerged by PESA-II. A UD-based comparison presented in Fig. 8, similar to the former comparison, shows no differentiation between NSGAII and NRGA while PESA-II has better distribution smoothness in collation with the other two. Fig. 9 considers their differentiation in solution diversity in terms of Di. As it is clear, the NRGA diversity is more than NSGA-II, and NSGA-II is more than PESA-II. The non-dominated solution for three MOEAs are exposed in Fig. 10 which shows that the performances of both NSGA-II and NSGA-II from the Pareto Front quality point of view are the same, howbeit they are weaker than PESA-II.

Fig. 7 Line plot and boxplot for RNI

Fig. 8 Line plot and boxplot for UD

Fig. 9 Line plot and boxplot for Di

Fig. 10 Line plot and boxplot for QM

A comparison between MOEAs by processing time is exposed in Fig. 6. As it is resolved, the NSGAII and NRGA processing times are almost equal and are twice that of PESA-II. The visualization in Fig. 7 compares the non-dominated result of all deployed MOEAs. As it is clear, the number of non-dominated ones for the NSGA-II and NRGA are almost the same

̃ is excellent) THEN RULE 1: IF (Time is free) and (QPF (SuitableSet is PESA-II.&.NSGA-II.&.NRGA) (0.2) ̃ is excellent) THEN RULE 2: IF (Time is free) and (QPF (SuitableSet is NSGA-II.&.NRGA) (0.8) ̃ is high) THEN RULE 3: IF (Time is free) and (QPF (SuitableSet is PESA-II.&.NSGA-II.&.NRGA) (0.3) ̃ is high) THEN RULE 4: IF (Time is free) and (QPF (SuitableSet is NSGA-II.&.NRGA) (0.3) ̃ is average) THEN RULE 5: IF (Time is free) and (QPF (SuitableSet is PESA-II.&.NSGA-II) (0.4) ̃ is average) THEN RULE 6: IF (Time is free) and (QPF (SuitableSet is PESA-II.&.NSGA-II.&.NRGA) (0.6) ̃ is excellent) RULE 7: IF (Time is bounded) and (QPF THEN (SuitableSet is PESA-II.&.NSGA-II.&.NRGA) (0.4) ̃ is excellent) RULE 8: IF (Time is bounded) and (QPF THEN (SuitableSet is NSGA-II.&.NRGA) (0.6) ̃ is high) THEN RULE 9: IF (Time is bounded) and (QPF (SuitableSet is PESA-II.&.NSGA-II.&.NRGA) (0.6) ̃ is high) THEN RULE 10: IF (Time is bounded) and (QPF (SuitableSet is NSGA-II.&.NRGA) (0.4) ̃ is average) THEN RULE 11: IF (Time is bounded) and (QPF (SuitableSet is PESA-II.&.NSGA-II) (0.8) ̃ is average) THEN RULE 12: IF (Time is bounded) and (QPF (SuitableSet is PESA-II.&.NSGA-II.&.NRGA) (0.2) ̃ is excellent) THEN RULE 13: IF (Time is exigent) and (QPF (SuitableSet is PESA-II.&.NSGA-II) (0.6) ̃ is excellent) THEN RULE 14: IF (Time is exigent) and (QPF (SuitableSet is PESA-II.&.NSGA-II.&.NRGA) (0.4) ̃ is high) THEN RULE 15: IF (Time is exigent) and (QPF (SuitableSet is PESA-II.&.NSGA-II) (0.7) ̃ is high) THEN RULE 16: IF (Time is exigent) and (QPF (SuitableSet is PESA-II.&.NSGA-II.&.NRGA) (0.3) RULE 17: IF (Time is exigent) and (QPF is average) THEN (SuitableSet is PESA-II.&.NSGA-II) (0.9) RULE 18: IF (Time is exigent) and (QPF is average) THEN (SuitableSet is PESA-II.&.NSGA-II.&.NRGA) (0.1) Fig. 11 Fuzzy rule base

Fig. 11 represents the final rule base for reasoning and making a policy on optimization. Sliding the line of each input variable (for finding the minded circumstance by the authors) poses the case that the Time should be almost bounded while the ̃ should be excellent. So the expected 𝑄𝑃𝐹 approximated reasoning infers that the suitable set of MOEAs that asserts all the MOEAs should be used. The final emerging Pareto Fronts according to the made policy are presented in Fig. 12. PESAII 0.93 0.92 0.91 0.9 0.89 0.88 0.87 0.86 0.85 1.78E+09

1.8E+09

NSGAII

NRGA

1.82E+09

1.84E+09

Fig. 12 Generated Pareto Front by MOEAs according to the policy

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