Multi-objective grey decision making

The main components of the traditional multi-objective grey decision model include the event set, strategy set, situation set, decision goal and decision weight.

First, construct the corresponding set of situations according to the event set and strategy set. Assume that A = {a₁,a₂⋯a_n} is the event set, B = {b₁,b₂⋯b_m} is the strategy set, s = {s_ij = (a_i,b_j)|a_i ∈ A,b_j ∈ B} is the situation set, and is the effect sample value of the situation under the target.

Second, select targets to determine the effectiveness measure: (1) The measure shown in Eq (1) is the upper effect measure and is mainly used to measure the degree to whichthe albino value deviated from the maximum whitening value.

(2)

The measure shown in Eq (2) is the lower effect measure and is mainly used to measure the degree to which the albino value deviated from the lower limit. (3) The measure shown in Eq (3) is the medium effect measure, where is the moderate value of the specified effect under the target.

Each of these three measures is applicable to different situations. The upper effect measure is used to optimize for larger values, whereas the lower effect measure can be chosen to minimize loss. The medium effect measure is used if the desired outcome of the effect is to be near a specified value.

Third, solve the consistent effect measure matrix for the situation set based on the effect measure of each target. (4) where is the consistent effect measure vector of situation s_ij under target k.

Fourth, establish the decision weight η_k(k = 1,2,⋯s), where , and solve the integrated effect measure r_ij and integrated effect measure matrix of situation s_ij.

(5)

(6)

Finally, if , then is the optimal strategy for event a_i; if , then is the optimal event for strategy b_j; if , then is the optimal situation.

Grey correlation analysis

Linear interpolation is used to transform the discrete behavior observations of the system factors into polylines of the segmented readings. Then, the model for the measure degree is constructed based on the geometric features of the polyline. The basic steps of the grey relational model are as follows:

Step1: specify the original sequence X₀(t) = {x₀(1),x₀(2), ⋯,x₀(n)} as the reference data sequence, also called the parent sequence, and X_i(t) = {x_i(1),x_i(2), ⋯,x_i(n)} as the sequence of data to be compared, also known as the subsequence.

Step2: make ξ_i(T) the correlation coefficient for sequences X₀(t) and X_i(t) at time T: (7) where 0.5 is the resolution factor and is typically between 0 and 1.

Step3: calculate the average of the correlation coefficients at each time of sequence X_i(t), i.e., the degree of correlation of subsequence X_i(t) to the parent sequence X₀(t): (8)

Evaluation of the target selection

The first step of the proposed weighting method is to select the evaluation target. Due to the variable effects of different EGR rates on the combustion and emissions of diesel engines, the selection of the evaluation indicators should consider the combustion and emission performance of the diesel engine as much as possible. In this paper, the fuel consumption rate, in-cylinder explosion pressure, NO_X, smoke and CO were selected as the evaluation targets. Because the main purpose of the EGR is to minimize the emission of NO_X pollutants, NO_X is defined as the main decision-making target, and the other four indicators are secondary decision-making targets.

The determination of the optimal EGR rate searches for the best compromise between the diesel engine combustion and emissions performances. The optimization of this compromise is reflected in the model as the primary consideration. Considering the important role of the target weights in the decision-making model, in the optimization, a compromise between diesel engine combustion and emission performance can be achieved by adjusting the target weight vector η_k(k = 1,2,3,4,5), where k values of 1, 2, 3, 4, and 5refers to the fuel consumption rate, cylinder burst pressure, NO_X, smoke and CO, respectively.

Establishment of the initial subjective weight vector

a) Assign the main decision-making target weight by considering the main purpose of the EGR, which is to effectively reduce the NO_X emissions. Therefore, expert scoring is first used to customize the target weight for the NO_X emissions based on the variable conditions of the diesel engine. The following rules were established by repeating the trials and reviewing the acquired information:

I. When the diesel engine is under low-load conditions (defined as≤50% load), the NO_X emission concentration is low. A lower EGR rate should be chosen to ensure the stability and economy of the diesel engine, thus making the NO_X weight η₃ = 0.4.

II. When the diesel engine is under high-load conditions (defined as ≥50% load), the NO_X emission concentration is high. A higher EGR rate should be chosen to ensure the necessary emissions, thus making the NO_X weight η₃ = 0.5.

b) After determining the weight of NO_X, the next step involves assigning the weights of the remaining indexes. Because each index represents a different aspects of diesel engine performance, relationships exist among indexes cannot be determined by direct observation or assessment. Therefore, grey relational analysis can be used as the basis for the distribution of weights.

The corresponding NO_X values (including the original machine value) at different EGR rates are used as the parent sequence, and the other four evaluation indicators corresponding to the value (including the original machine value) are used as subsequences. Then, the correlation coefficient between the other four evaluation indexes and the NO_X index is solved by grey relational analysis. Finally, the correlation coefficient between the primary and secondary decision goals can be obtained as (9)

The correlation coefficient represents the degree of proximity of the secondary decision objective to the primary decision objective, which reflects their relative importance to the main decision-making targets.

c) Solve the initial subjective weight vector, defined as η₃ and r_i. The other four decision-making target weight values η_k(k = 1,2,4,5) are determined using the formula r_i(1-η₃), and the initial subjective weight vector is ascertained.

Establishment of the objective weight vector

The objective weight vector is based on the theory of uncertain decision-making combined with the characteristics of the EGR test data. Grey entropy analysis is introduced to explore the intrinsic relationship between each of the decision objectives, and the objective weight value is obtained.

In this paper, the event set A = {a₁}, i.e., the event is the best EGR rate decision. The strategy set B = {b₁,b₂⋯b_m} consists of m decision-making programs, and b_m represent different EGR rates.

According to the consistent effect measure matrix of each situation set for target k, is defined as the positive ideal measure that represents the best test value for each index under current operating conditions, and is defined as the negative ideal measure of target k, that represents the worst test value for each index under current operating conditions.

The deviations of the consistent effect measure with its corresponding positive and negative ideal effect measures and , respectively, are (10)

For event a₁, the total deviation of the consistent effect under each strategy and target with its corresponding positive and negative ideal effect measures and , respectively, can be expressed as a function of the target weight as (11)

Next, the weight η_k is determined such that the total deviation of the positive and consistent effect measures under each situation is minimized. The total deviation of the negative and consistent effect measures under each situation is maximized, which can be attributed to the following multi-objective optimization problem: (12)

Due to the uncertainty of the weight from the decision-making system with incomplete information, the uncertainty of the weight sequence should be minimized when determining the target weight. The target weight sequence is a grey connotation sequence η_k(k = 1,2,⋯,s), and its grey entropy can be defined as (13)

According to the principle of maximum entropy, the determination of the target weight should minimize the uncertainty of the weight sequence. Therefore, the maximization constraint of the grey entropy is applied: (14)

Then, and can be transformed into a single target optimization problem: (15) where μ represents the balance factor between the three objectives and μ is typically 1/3.

Then, construct a Lagrangian function: (16)

Finally, η_k can be solved as (17)

Establishment of the subjective and objective comprehensive weight vector

The objective weight vector η_k(k = 1,2,3,4,5) and the initial subjective weight vector η_0k(k = 1,2,3,4,5) are combined to obtain the subjective and objective comprehensive weight vector η_1k(k = 1,2,3,4,5), which is typically obtained by the following formula: (18)

Acquisition of test data

A model of a specific turbocharged diesel engine is used as the research object to verify the effectiveness of the optimization method. The main technical parameters of the diesel engine are provided in Table 1.

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Table 1. Main technical parameters of the TBD234V12.

https://doi.org/10.1371/journal.pone.0191626.t001

The test included the low, medium and high three-speed test, where the three speeds are 25%, 50%, and 75% load, respectively, under a total of 9 working conditions. A portion of the operating point test data is shown in Table 2. cgr, fc, co, no, soot and cbp represent the EGR rate, fuel consumption, CO, NO_X, soot and cylinder burst pressure, respectively.

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Table 2. A portion of the operating point test data.

https://doi.org/10.1371/journal.pone.0191626.t002

Analysis of the results

I.Low-load conditions

OP1 and OP2 represent the low speed at 50% load and medium speed at 25% load, respectively. Taking OP1 as an example, the EGR rates were 2.2%,4.6%,7.5%,9.8% and 11.5%. The effect sample matrix is based on experimental data under different EGR rates:

Among these values, the abscissa represents the fuel consumption rate, CO, NO_X, soot and in-cylinder burst pressure, respectively. The ordinate represents different EGR rates; for example, the first column represents an EGR rate of 2.2%.

Step 1: solve the consistent effect measure matrix:

Step 2: solve the initial subjective weight vector; OP1 belongs to the low-load operating point, η₃ = 0.4. Determine the grey association sequence:

Mother sequence: Subsequence:

The correlation coefficient between the other four evaluation indexes and the NO_X index are r_i = [0.2845 0.2915 0.2067 0.2173]. The initial subjective weight vector is η_k = [0.1707 0.1749 0.4000 0.1240 0.1304].

Step 3: solve the comprehensive weight vector. The objective weight vector is obtained through entropy analysis as η_0k = [0.1872 0.1816 0.1596 0.2445 0.2270]. The integrated weight vector is η_k = [0.1704 0.1694 0.3405 0.1617 0.1579].

Step 4: Solve the comprehensive effect measure matrix and sort the advantages and disadvantages according to the optimal principle, R = [0.9052 0.9027 0.9031 0.8735 0.8203]. The following performance ranking of the different EGR rates is obtained for OP1:2.2%>7.5%>4.6%>9.8%>11.5%. The optimal EGR rate is 2.2% for this condition, and when the EGR rate is less than approximately 8%, there is only a slight difference between the comprehensive evaluation values obtained with the different EGR rates. When the EGR rate increases, the comprehensive performance evaluation value decreases significantly. Therefore, a smaller EGR rate should be adopted.

The performance ranking of different EGR rates under OP2 and OP3can be obtained in a similar manner as follows:

The optimal EGR rates of theOP2 and OP3 conditions are 4.5% and 2.2%, respectively. When the EGR rate rises, the corresponding evaluation value decreases, and the decline is more significant when the EGR rate is greater than approximately 8%.

The results of OP1, OP2 and OP3 indicate that a lower EGR rate should be adopted when the diesel engine is under low-load conditions. The comprehensive evaluation value significantly decreased with increases in the EGR rate. The analysis of the reasons for this result indicated that when at low load, the NO_X pollutant emissions is low and a portion of the dynamic performance of diesel engines will be consumed if the EGR rate is excessively high. The EGR rate must be reduced to ensure the economy and power of diesel engines.

II.High-load conditions

OP4, OP5 and OP6 represent a 75% load at different speeds. Let η₃ = 0.5; then, based on the simulation calculations, the comprehensive effect measure matrices of OP4, OP5 and OP6 are as follows:

The performance ranking of the different EGR rates under OP3 and OP4 can be obtained as

The results for OP4, OP5 and OP6 indicate that the comprehensive evaluation value of each EGR rate increases with increases in the EGR rate, and the change is significant when the EGR rate is greater than 8%. This result indicates that a lower EGR rate is less effective at improving the overall performance of diesel engines when the diesel engine is working at a high load. Furthermore, according to the results for OP6, when the EGR rate increases to 11.1%, the corresponding comprehensive evaluation value decreases. This trend occurs because OP6 is at high speed and high-load conditions to ensure adequate power performance; thus, a high EGR rate should not be used.

In summary, the above assessment and decision-making results demonstrate that when working under low-load conditions, because of the lower NO_X emission concentration, a smaller EGR rate should be used to balance the power and economy of diesel engines. When working under high-load conditions, the NO_X emission concentration is high and a higher EGR rate should be adopted to ensure the emissions performance. However, excessive EGR rates may have a negative impact on diesel engines when working under high-speed, high-load conditions. This finding is consistent with the characteristics of EGR performance for the current turbocharged diesel engine and demonstrates the effectiveness of the optimization method. However, compared to existing methods, the proposed method has a higher modeling and simulation efficiency and provides more reasonable results. Therefore, it can replace existing methods as a more convenient approach for engineering applications.

The EGR rate is limited to less than 15% in this study, and no further data can be obtained. However, the authors believe that this limit does not affect the main concept of this method because the core of this method is concerned with the EGR performance evaluation and decision-making problem with less data and more uncertainty, which is also the characteristic of this method that distinguishes it from the existing evaluation methods. Therefore, this method can provide effective theoretical support for the current EGR performance evaluation and decision-making of turbocharged diesel engines, which has certain guiding significance.