Mingtao DONG,Jianhua CHENG?,Lin ZHAO

College of Intelligent Systems Science and Engineering,Harbin Engineering University,Harbin 150001,China

Abstract:This paper proposes a combination weighting (CW) model based on iMOEA/D-DE (i.e.,improved multiobjective evolutionary algorithm based on decomposition with differential evolution) with the aim to accurately compute the weight of evaluation methods.Multi-expert weight considers only subjective weights,leading to poor objectivity.To overcome this shortcoming,a multiobjective optimization model of CW based on improved game theory is proposed while considering the uncertainty of combination coefficients.An improved mutation operator is introduced to improve the convergence speed,and thus better optimization results are obtained.Meanwhile,an adaptive mutation constant and crossover probability constant with self-learning ability are proposed to improve the robustness of MOEA/D-DE.Since the existing weight evaluation approaches cannot evaluate weights separately,a new weight evaluation approach based on relative entropy is presented.Taking the evaluation method of integrated navigation systems as an example,certain experiments are carried out.It is proved that the proposed algorithm is effective and has excellent performance.

Key words:Combination weighting;MOEA/D-DE;Game theory;Self-learning ability;Relative entropy

1 Introduction

The shortcomings of traditional testing methods,i.e.,sometimes testing is impossible to carry out and the subjective indices cannot be evaluated,can be overcome by the evaluation method applied to inte‐grated navigation systems.Therefore,this field has attracted increasing attention from scholars (Wang XL et al.,2017).The evaluation results can discern the differences between various systems,thereby provid‐ing technical support for decision makers and offer‐ing guidance for system optimization.The integratednavigation system contains a lot of indices with dif‐ferent attributes and multi-level characteristics.Indi‐ces with complex features put forward higher require‐ments for evaluation methods.The means to widen the application of the evaluation method to an inte‐grated navigation system requires further studies.

Combination weighting (CW) is one of the key tools of evaluation methods,and the accuracy of CW directly affects the evaluation result.Scholars have carried out a lot of research on CW,and the solutions include mainly the direct approach (Jiao et al.,2016;Cheng et al.,2019a;Zhu et al.,2019;Lv et al.,2020;Pan et al.,2020),single objective model approach(Xu and Cai,2012;Liu and Hu,2015;Dai and Niu,2017;Zhou RX et al.,2017;Zhang ZC and Chen,2018),multiobjective model approach (Shi et al.,2012;Yin et al.,2016;Cheng et al.,2019b),and CW approach based on game theory (GT) (Lai et al.,2015;Sun LJ et al.,2016;Li AH,2017;Yang et al.,2018;Dong et al.,2020).

In previous studies,the improved analytic hier‐archy process (AHP) was employed to compute the subjective weight(SW),the enhanced entropy method(EM)was used to compute the objective weight (OW),and the weighted sum approach (WSA) was applied to establish the CW model.The combination coeffi‐cients (CCs) were set according to the preference of decision makers (Pan et al.,2020).SW was computed by AHP,the correlation method was used to compute the OW,and two different weights were combined by the WSA.CCs were determined by the variation coefficient of each weight based on AHP (Lv et al.,2020).Generally,SW and OW were computed by AHP and EM,respectively.WSA was used to com‐bine different weights,and the difference between various methods is in the solutions of CC (Jiao et al.,2016;Cheng et al.,2019a;Zhu et al.,2019).The in‐fluence of CC on evaluation results was analyzed,and CCs were selected by subsequent results (Jiao et al.,2016).CCs were directly set in some studies(Cheng et al.,2019a;Zhu et al.,2019).In the above literature,WSA was employed to establish the CW model based on SW and OW.The solutions of CCs are simple and have low precision.

The direct approach has poor accuracy,and there‐fore the single objective model approach was proposed to solve the problem.To reflect the consistency of subjective and objective information,where SW was obtained by matter-element analysis and OW was ac‐quired by EM,a single objective model was present‐ed by combining index values with weights,and CCs were obtained by solving the model (Dai and Niu,2017).In a different approach,SW was comput‐ed by AHP,and OW was computed by the rough EM.A single objective model was established based on the minimum deviation,and CCs were computed by the cooperative game method (Zhang ZC and Chen,2018).Alternatively,OW was obtained by the rough set method,and SW was derived by AHP.Combined with the scheme value,a linear objective model was proposed based on the principle of maxi‐mizing variance.Therein,CCs were computed by the Lagrange multiplier method (LMM) (Liu and Hu,2015).For the multi-expert weight,the weight‐ed arithmetic average operator was used to trans‐form weights into CW.From the perspective of mini‐mizing group disharmony,a nonlinear optimization model based on deviation function was established,and the model was computed by the genetic algo‐rithm(Xu and Cai,2012).In a further study,SW and OW were obtained by fuzzy AHP and the projection pursuit method,respectively.The single objective model was established based on the principle of the mini‐mum relative entropy,and the model was computed by LMM (Zhou RX et al.,2017).The above literature shows that the single objective model approach must use the index values,and that the model is most com‐monly computed by LMM.The accuracy of the index depends on the dimensionless method.So,the accu‐racy of the single objective model approach is low,and it is not suitable for the complex optimization model.

A double-objective model of CW was subse‐quently proposed,which was transformed into a singleobjective optimization model by WSA.LMM was employed to compute CCs(Yin et al.,2016).Consid‐ering the randomness of the weight itself and the con‐sistency between weight vectors,a nonlinear multi?objective model of CW was also established.The model was transformed into a single objective model by WSA.An improved particle swarm optimization(PSO) algorithm was proposed to compute CCs (Shi et al.,2012).A different multiobjective model was es‐tablished by expert weights,and the model was based on the uncertainty of CCs and the consistency of weights.WSA was used to transform the model into a single objective model,and the modified differential evolution (DE) algorithm was applied to compute CCs(Cheng et al.,2019b).When the multiobjective model is transformed into a single-objective model,it is diffi‐cult to obtain model coefficients accurately.This is a shortcoming of the multiobjective model approach.In comparison with the single-objective model,the multi‐objective model of CW can express the relationship between different weights in a comprehensive way.

The CW model based on GT was further estab‐lished,and the condition of the optimal first order was used to compute CCs based on the differential prop‐erty of matrix (Lai et al.,2015).Furthermore,fuzzy AHP and EM were used to compute SW and OW respectively,and the CW model was established based on GT.Therein,CCs were computed by the dif‐ferential property of matrix (Sun LJ et al.,2016).In another model,SW and OW were computed by AHP and EM,respectively.Two weights were combined into CW based on GT,and CCs were computed based on the differential property of matrix (Yang et al.,2018).Aiming at the problem that traditional CW based on GT may obtain negative weights,an im‐proved combination weighting method based on GT(ICWGT) was proposed,where CCs were computed by LMM (Li AH,2017).Dong et al.(2020) obtained CW by the idea of GT,which was combined with SW and OW.SW was obtained by interval AHP,and OW was acquired by information entropy.The condi‐tion of the optimal first order was used to compute CCs.The above studies demonstrated that the GTbased CW approach is a new idea in CW,which is essentially a single-objective model approach.Based on the above literature,it is clear that the uncertainty of CCs in a CW model should be considered.

In the past decades,many intelligent algorithms have been applied to multiobjective optimization problems.Many scholars have focused on intelligent algorithms for a multiobjective optimization model of CW (Shi et al.,2012;Cheng et al.,2019b).The multiobjective optimization problem was transformed into several scalar optimization subproblems.The sub‐problems were optimized at the same time,which could generate a highly uniform distribution of solu‐tions.This method was named multiobjective evolu‐tionary algorithm based on decomposition(MOEA/D)(Zhang QF and Li,2007).Li H and Zhang (2009)proposed the further developed MOEA/D-DE,a com‐bination of MOEA/D and DE algorithm,which has the ability to deal with the complex Pareto front(PF).Simulation results showed that this new algo‐rithm is better than MOEA/D.The MOEA/D and DE algorithm subsequently became the most popular of these new intelligent algorithms.

When a multiobjective optimization model is nonlinear and contains an equality constraint,the convergence of the DE algorithm is insufficient.Das and Suganthan (2011) reviewed the basic concepts of discrete evolutionary algorithm and presented its appli‐cations for multiobjective,constrained,large-scale,and uncertain optimization problems.Ye et al.(2013)summarized existing problems of the DE algorithm,and reported the main issue that it is necessary to choose suitable parameters to ensure the success of the algorithm.Al-Dabbagh et al.(2018) pointed out that the performance of the DE algorithm dependeds on the control parameters which determine the solution quality and search efficiency.Current studies have focused on how to optimize the control parameters.Zhang CM et al.(2014),for example,proposed an adaptive adjustment strategy ofFand CR,whereFrepresents the mutation control parameter and CR represents the crossover control parameter,and the parameters were obtained adaptively by the fitness value.Moreover,Fan and Yan (2016) presented an adaptive DE algorithm,which was used to control the partition evolution of parameters and the adaptive mutation strategy.The mutation strategy was auto‐matically adjusted with the population evolution,and parameters were evolved in their own partition to find the near-optimal value adaptively.Li YZ et al.(2020)presented an improved DE algorithm based on a dual mutation strategy to reduce the influence of the muta‐tion strategy and parameter selection on the DE algo‐rithm.Wu et al.(2013) introduced both an improved mutation operator and a parameter adaptive strate‐gy in the DE algorithm.Ding et al.(2020) described an adaptive strategy that can adaptively select the dif‐ferential mutation operator with the population evolu‐tion.In the above literature,the latest achievements in MOEA/D and DE algorithm were summarized.The CW model with nonlinear characteristics contains an equality constraint,which puts forward higher require‐ments for the robustness of intelligent algorithms.

The solutions of CW were systematically desc?ribed in the literature,and their advantages and disad‐vantages were subsequently analyzed.Each of them may provide a new idea for solving the multiobjec‐tive optimization problem of CW.The purpose of this paper is to propose a multiobjective optimization model of CW based on improved MOEA/D with dif‐ferential evolution(iMOEA/D-DE).First,a multi?objective optimization model of CW based on improved GT is developed to overcome the poor objectivity of the multi-expert weight.Second,due to the CW model being nonlinear and containing an equality constraint,MOEA/D-DE has the disadvantage of poor convergence.Therefore,inspired by Wu et al.(2013),an improved mutation operator is intro‐duced,and a new adaptive mutation constant and crossover probability constant are proposed.Finally,a new weight evaluation approach is presented to evaluate CW.

2 Multiobjective optimization model of CW

Cheng et al.(2019b) proposed a multiobjective optimization model of the multi-expert weight,which is based on the uncertainty of CCs and the weight consistency.The multi-expert weight was composed of expert weights,and had poor objectivity.A doubleobjective constrained optimization model was devel‐oped to overcome these drawbacks,and an improved adaptive penalty function was described to handle the constrained problem,but the penalty coefficient was difficult to obtain accurately (Cheng et al.,2021).In this paper,an improved GT is introduced and a new multiobjective model of CW is proposed to address this limitation.The equality constraint included in the CW model makes the solution more complex,and thus the multiobjective optimization method is introduced to deal with this constraint.

2.1 Multiobjective optimization model

2.1.1 CW model based on improved GT

The CW model based on GT aims to seek a bal‐ance between the weights of different weight methods,so as to minimize the deviation between CW and each weight.Suppose thatnweighting methods are used to compute the weight value.A weighting method con‐tains the SW method (Cheng et al.,2019a) and the OW method (Dai and Niu,2017).Each weight iswi=(wi1,wi2,…,wim),i=1,2,…,n,andmis the number of indices.Let CCs of CW bek1,k2,…,kn.Then CW can be obtained by WSA:

To minimize the deviation between CW and each weight,the CW model based on GT is established:

where||·||2is the 2-norm of a matrix.

According to the differential property of the ma‐trix(Lai et al.,2015),the optimal condition of Eq.(2)is

A set of solutions{k1,k2,…,kn}can be obtained by solving Eq.(3).After normalization,can be obtained.Subsequently,CW is

CW depends on Eq.(4),while the result of this formula depends on Eq.(3).Therein,negative results may be obtained,resulting in a negative weight.There‐fore,a CW model based on improved GT is intro‐duced(Li AH,2017).The new CW model is

where|·|is the absolute value andk=(k1,k2,…,kn).

The constraint of Eq.(5)is

wherenindicates the number of weights.

2.1.2 Uncertainty of CW

Based on mathematical statistics,the real weight value of each index is a random value.The weight values of different weighting methods are sample values of the true weight.Different weighting methods bring uncertainty to CW.The Shannon information entropy is usually used to describe this uncertainty(Shi et al.,2012).The uncertainty of CW is (Cheng et al.,2019b)

Since 0≤ki≤1 and the objective functionf2con‐tains “l(fā)n” function,the minimum value of CC cannot be taken as 0,but is assumed to be 0.01.The con‐straint of the uncertain model is the same as in Eq.(6),but the value range ofkibecomes 0.01≤ki≤1.

2.1.3 A new multiobjective optimization model of CW

Considering that CW is based on improved GT and the uncertainty of CW,the multiobjective optimi‐zation model of CW is proposed as

whereF(k)is the objective function.

Eq.(7) is a maximum optimization model,while model(5)is a minimum optimization model.Two mod‐els with different characteristics cannot be linked together to form a multiobjective optimization model.Hence,model (7) is converted into a minimum model.Next,it is combined with Eq.(5) to obtain Eq.(8).Eq.(9) is the common constraint off1andf2,and the value range ofkiis 0.01≤ki≤1.

2.2 CW based on the multiobjective optimization method

The constraints in the multiobjective optimiza‐tion model make the solving process rather difficult.Wang Y et al.(2009) summarized the progress of constraint processing technology in the optimization model.Among the solutions,the multiobjective opti‐mization method transforms the constraint into objec‐tive functions,and these functions are treated as dif‐ferent objective functions,which can overcome the drawback of difficulty to accurately obtain the pen‐alty coefficient.In this study,the multiobjective opti‐mization method is used to deal with the constraint.

The CW model contains an equality constraint,which is usually transformed into an inequality con‐straint(Wang Y et al.,2009):

whereδis a tolerance value of equality constraint,and is generally a small positive number of 0.001 or 0.0001.In general,the degree to which the CC violates the equality constraint can be expressed as

wheref3(k) is the infeasibility degree ofk.Through combining Eq.(11) with Eq.(8),the multiobjective optimization method is used to transform them into a multiobjective optimization model with three objec‐tive functions.Therefore,the three-objective optimi‐zation model of CW is formulated as

wherek=(k1,k2,…,kn),andf3is called the feasible solution.

3 CW model based on iMOEA/D-DE

In Section 2,the multiobjective optimization model of CW is presented.The CW model is nonlinear and contains an equality constraint,which makes it difficult to be solved.The MOEA/D-DE (Li H and Zhang,2009),an improved algorithm of MOEA/D,is capable of dealing with the problem with complex PF.The CW model puts forward a higher requirement for the dis‐tribution and convergence of the MOEA/D-DE algo‐rithm,and thus an iMOEA/D-DE algorithm is proposed.

3.1 iMOEA/D-DE

3.1.1 Classical MOEA/D-DE

The main idea of MOEA/D is to break down the multiobjective optimization model into several scalar subproblems,and use their neighborhood information problems to optimize all subproblems simultaneously.The DE operator and polynomial mutation operator are used to generate new solutions.The important for‐mulas of MOEA/D-DE are as follows:

1.Use the Tchebycheff approach to aggregate function values,which are obtained by decomposition operation:

wherez*is the reference point,=min{fi(k) },the weight vector satisfiesλ=(λ1,λ2,…,λm),λi≥0,andm=3 represents the number of objective functions.

2.Determine the renewal range and produce new individuals:

whereδ1denotes the probability that parent solutions are selected from the neighborhood,rand is a random mumber within [0,1],Nis the population size,andB(i)={i1,i2,…,iT},and,λi1,λi2,…,λiTare the nearestTvectors ofλi.

The DE operator and polynomial mutation oper‐ator are used to generate new individuals:

In Eq.(15),CR denotes the crossover probability with a value range of 0–1,andFis a scalar number with a value range of 0?1.Letr1=i,randomly selectr2andr3fromP,and use the DE operator to generateis thekthelement inyˉ.Three individualsandare selected from their parents;is thekthele‐ment ofxi.In Eqs.(16) and (17),pmis the mutation rate,ηstands for the distribution index,andakandbkare the lower and upper bounds of thekthdecision variable,respectively.

3.1.2 Improved mutation operation

The DE/rand/1 operator is used in the differen‐tial mutation stage of MOEA/D-DE (Li YZ et al.,2020),in which the operator is beneficial to global search,but not conducive to improving the conver‐gence rate of the algorithm.Various operators have different characteristics and are suitable for differ‐ent problems.In the CW model,the existence of equality constraint leads to a very small proportion of the feasible region in the search space and a large degree of dispersion,and the constraint re‐quires the operator to have a better global and con‐vergence rate.Therefore,the improved mutation operation is introduced in the differential mutation stage(Wu et al.,2013).

Based on the DE/rand/1 operator,the idea of im‐proved mutation operation is to sort three individuals by their fitness values,which are randomly selected from their parents.Assume that the order isxa,G,xb,G,andxc,G.Taking the optimalxa,Gas the mutation refer‐ence vector and the difference between the suboptimal and the worst individual (xb,G?xc,G) as the difference vector,the improved mutation operation is obtained as follows:

It is known that the improved mutation opera‐tion takesxb,G?xc,Gas the difference vector,which is beneficial to improve the convergence rate.Hence,Eq.(15)can be updated to

3.1.3 Adaptive strategy with self-learning ability

In MOEA/D-DE,Fand CR are both constant values in the DE operator,and remain the same during the evolution process.They have an important impact on the performance of the algorithm,and parameter selection is usually associated with the problem.The CW model with the equality constraint is nonlinear,which makes the model more difficult to be solved,and the constant values ofFand CR are hard to adapt.Taking the adaptive idea from the literature(Wu et al.,2013),a new adaptive strategy is proposed.According to the state of population evolution,an adaptive strat‐egy with self-learning ability is formulated,which is independent of the optimization problem.As shown in Fig.1,Fand CR are coded simultaneously with the individuals.In each generation,xi,Ghas a correspond‐ingFi,Gand CRi,Gvalue.The initial values ofFand CR are randomly selected within the scope of their respective values.In the process of evolution,if a better individual cannot be produced in five genera‐tions,it indicates that the relevant parameters are not suitable and need to be reset.If one or more better individuals are produced within five generations,the relevant parameters should be retained.The parameter with the most reserved times is the most appropriate parameter.Obviously,inappropriate parameters are constantly reset,and the ultimate goal is the most appro?priate parameter that is consistent with the basic idea of the evolutionary algorithm.

Fig.1 Adaptive coding format

Based on the above idea,a parameter adaptive strategy with self-learning ability is proposed:

wherecrepresents the number of optimal individuals generated in the five generations,FmaxandFminare the maximum and minimum values ofFrespectively,and CRmaxand CRminare the maximum and minimum values of CR respectively.

In the CW model,the existence of equality con‐straint makes the feasible region more discrete,which requires better diversity in the initial stage of evolu‐tion.With regards toF,to maintain the diversity of individuals,Fis required to decrease fromFmax.As the algorithm iterates,Fis required to decrease to avoid damage to the optimal solution to retain valu‐able information.The decrease amplitude is the value ofFmax?Fmin.

The value of CR determines whether the muta‐tion vector or the target vector is used in the cross‐over operation (Zhang CM et al.,2014).When a better individual is produced,the test vector should take the variation vector with a greater probability;thus CR should be taken at a larger value.On the contrary,if the test vector takes the target vector with higher probability,CR should take a smaller value.In the early evolution,the feasible solutions are widely dis‐tributed and there are bad individuals,and hence CR should be taken at a smaller value.Therefore,CR increases gradually from CRmin,and the decrease amplitude is the value of CRmax?CRmin.

In the process of evolution,if a better individual cannot be produced within five generations,thencis equal to 0 and a newFis produced using rand until a better individual is produced.As the algorithm iter‐ates,Fand CR are gradually changed towards the best solution.The complexity of the CW model makesFand CR change dynamically.Thus,Fand CR have self-learning ability.

3.1.4 Flow of iMOEA/D-DE

The MOEA/D-DE is combined with improved mutation operation and the adaptive strategy with self-learning ability to obtain the iMOEA/D-DE algorithm.

Step 1:The population size isN,the number of neighborhood weight vectors of each weight vector isT,the probability of selecting parent individuals from neighborhood isδ1,the child solution number replaced isnr,and the maximum number of iterations isGmax.

Step 1.1:Fori=1,2,…,N,setB(i)={i1,i2,…,iT},whereλi1,λi2,…,λiTare the nearestTweight vectors ofλi.

Step 1.2:Generate the initial populationPwith scaleNrandomly,assuming that FVi=F(xi).The ini‐tialization reference point isz=(z1,z2,…,zm),wheremis the number of objective functions.

Step 2:According to Eq.(14),select the update matching/updating range.

Step 2.1:Suppose thatr1=iand that two indicesr2andr3are randomly selected fromP.Three individ‐uals are then sorted bygteto obtain the variation reference and difference vectors.

Step 2.2:Judge whether the new optimal indi‐viduals are generated within the five generations,that is,whether the parametercis greater than zero.Fand CR are adaptively adjusted by Eqs.(20) and (21),respectively.Ifcis greater than zero,it is assigned zero value.According to DE and the polynomial muta‐tion operator,a new solutionyis generated by param‐eterpm.

Step 2.3:Judgey,and if it exceeds the feasible region,it will be randomly selected in the feasible region.Ifzj>fi(y),setzj=fi(y),j=1,2,…,m.

Step 2.4:Suppose thatca=0.

(1) Ifca=nrorPis an empty set,then turn to step 3;otherwise,randomly select indexjfromP;

(3)RemovejfromPand proceed to step 1.

Step 3:Judge whether the program termination condition is met;if it is not,go to step 2.

Step 4:Output the objective function value and the optimal solution set.

3.2 A new weight evaluation approach

To evaluate the rationality of the CW model,the weight evaluation approach must be used.Existing weight evaluation approaches,however,have some drawbacks:(1) Using the weight value and the index value to compute the deviation function,the result is affected by the accuracy of the index conversion method (Shi et al.,2012);(2) The weight evaluation approach and scheme ranking are linked together(Song and Yang,2004),which is not applicable to the problem with the weight only.Hence,it is necessary to propose a new weight evaluation approach for CW.

CW combines SW and OW.When CW is obtai?ned,CW and each weight are independent,and can be regarded as independent discrete distributions.From the perspective of CW and each weight,the deviation between two distributions should not be too large and should tend to be consistent;the relative entropy can represent the deviation between two distributions.Definition 1(Zhou YF and Wei,2006) Assuming thatandis called the relative entropy ofxrelative toy,wherex=(x1,x2,…,xn)andy=(y1,y2,…,yn).Its main properties are as follows:

(2) The necessary and sufficient condition of=0 is thatxi=yifor alli.

Based on the above properties,whenxandyare two discrete distributions,the relative entropy can be used as a measure of their coincidence.The weight eva?luation measure of CW can then be obtained as follows:

whereW=(W1,W2,…,Wn) is the CW vector andw=(w1,w2,…,wm) is the weight vector,and the relation‐ship betweenWandwis discussed in Eq.(1).The smaller the relative entropy,the smaller the differ‐ence between CW and each weight.

4 Results and discussion

Experiments are carried out to verify the effec‐tiveness of the proposed approach.First,to verify the performance of iMOEA/D-DE,two kinds of test instances are used for comparison with MOEA/D-DE(Li H and Zhang,2009).Second,to verify the perfor‐mance of the proposed algorithm in the CW model,the iMOEA/D-DE is compared with MOEA/D (Zhang QF and Li,2007),MOEA/D-DE,and NSGA-II (Deb et al.,2002).Finally,the weight evaluation approach is verified in the CW model.

4.1 Test instance experiments

In this study,two kinds of test instances with different properties are selected by the characteris‐tics of CW,where the number of the objective func‐tions is 2 and 3,separately.The first is a two-objective test instance ZDT (Zhang QF and Li,2007) and the second group is a three-objective test instance DTLZ(Zhang QF and Li,2007),which are used to test the PF.

The MOEA/D-DE is compared with the iMOEA/D-DE.To ensure the comparability of test results,the same parameters are set in both algorithms,and readers can refer to Li H and Zhang (2009) for parameter selection.The parameters are set as follows:the value range ofFis [0,1],and its initial value is set to 0.5;the value range of CR is [0,1],and its initial value is set to 0.5;the distribution index ofηis 20;the poly‐nomial variation ratepmis 1/n(nis the number of decision variables);the number of neighborhoodsTis 0.1N;the probability of selecting parents from neigh‐borhoodδis 0.9;the child solution number replacednris 2;the population size of ZDT is 100;the popula‐tion size of DTLZ is 300.

To evaluate the distribution and convergence of the algorithm,the inverted generational distance(IGD) (Ishibuchi et al.,2018;Sun YN et al.,2019)is introduced.The smaller the IGD,the better the distribution and convergence of the PF.IGD requi?res the target object to obtain real PF.Two kinds of test instances meet the requirements,and IGD is as follows:

wherep*is a set of reference points for IGD,PF indi‐cates the nondominated solutions generated by the al‐gorithm,and dist(p,PF) is the nearest distance frompto PF.

Table 1 lists the results of the two algorithms in ZDT and DTLZ.MOP is the multiobjective optimiza‐tion problem.Each test instance runs 20 times to com‐pute the mean value,standard deviation(STD),and the minimum value (min) of IGD.The bold font indicates that the values represent excellent performances.The iMOEA/D-DE is better than the MOEA/D-DE with respect to the standard deviation of IGD,and the per‐formances of these two algorithms are equivalent to the minimum value of IGD.However,iMOEA/D-DE is worse than MOEA/D-DE in the mean value of IGD,because the middle point of iMOEA/D-DE for the mean value of IGD is smaller than that of MOEA/D-DE.Hence,the performances of these two algorithmsare similar,indicating that the iMOEA/D-DE algorithm is effective.

Table 1 IGD of ZDT and DTLZ

4.2 CW model experiments

For the integrated navigation system,OW de‐pends on the simulation data.Taking the accuracy in‐dex in the index layer in the literature (Cheng et al.,2019b) as an example,the weights of the first three experts are (0.3,0.2,0.5),(0.2,0.3,0.5),and (0.2,0.4,0.4).According to the simulation results of the integrated navigation system in Table 2,the objective weights are (0.3148,0.3431,0.3421) by EM (Jiao et al.,2016).OW and SW are substituted into Eq.(12)to obtain the three-objective optimization model of the CW.To verify the effectiveness of iMOEA/D-DE in the CW model,it is compared with MOEA/D,MOEA/D-DE,and NSGA-II.The parameters of the four algorithms refer to the parameters in Section 4.1 and the parameters in each reference.

Since the multiobjective model of CW is unable to obtain real PF,IGD cannot be used to measure the performance of the algorithm.The hyper-volume(HV) (Tian et al.,2016;Cai et al.,2021) is used to measure the distribution and convergence of the algo‐rithm,and is described as

wherePis the optimal solution set,and the reference point isr=(r1,r2,…,rm).HV is designed to find an area sum or volume sum of optimal solution relative to the reference point.MOEA/D is used to plot the figure of the CW.According to the CW distribution in Fig.2,the reference point is set asr=(0,1.3,0.5).The larger the HV,the better the distribution and con‐vergence of the algorithm.

Table 2 Results of the integrated navigation system

In Table 3,HVs of four algorithms are given.The order of HVs from large to small is:iMOEA/DDE,NSGA-II,MOEA/D-DE,and MOEA/D.The HV of iMOEA/D-DE is the largest,which indicates that the proposed algorithm has the best distribution and convergence performance.

4.3 Multi-expert weight and CW experiments

Fig.2 Figure of the CW model based on MOEA/D

Table 3 HV values of the four algorithms

Based on the literature (Cheng et al.,2019b),the multi-expert weight of the integrated navigation system is shown in Table 4,and the test data calcu‐lated by CW is shown in Table 2.The CW combines three SWs with one OW.The three weights are ex‐pert 1,expert 2,and expert 3.DandCin Table 4 in‐dicate two index layers.Dindicates the device layer.Cincludes the accuracy,stability,and usability in the index layer.OW is computed by EM from the data in Table 2.The results are shown in Table 5.Since there are no test data for indices in the function layer,there is only the multi-expert weight.Four weights are sub‐stituted into Eq.(12),and CWs are subsequently computed by iMOEA/D-DE.The results are shown in Table 6.

To compare the multi-expert weight with CW,the comparison curves of two weights are plotted,as shown in Fig.3.The difference between the multi-expert weight and CW is whether the fourth weight is SW or OW.It can be seen from Fig.3 that the existence of OW changes the size of each CW component.Each weight of the multi-expert weight is SW without OW,while the CW includes both SW and OW.The objec‐tivity of weight is reflected in the source of data.The data of OW comes from the index attribute value,while the data of SW comes from experts,which has the advantage of strong explanation.CW is objectiveand interpretable at the same time,thus overcoming the drawback of multi-expert weight.Therefore,CW is more reasonable and objective than the multi-expert weight.

Table 4 Weights of the three experts

Table 5 Objective weight(OW)of the integrated navigation system

Table 6 Combination coefficient(CC)and combination weighting(CW)

Fig.3 Comparison of the multi-expert weight and CW:(a) weights of the device layer;(b) weights of precision index in the index layer;(c)weights of stability index in the index layer;(d)weights of usability index in the index layer

4.4 Experiments with the new weight evaluation approach of the multi-expert weight and CW

To verify the new weight evaluation approach,experiments are carried out with examples in the lit‐erature (Shi et al.,2012).Compared with the results of the two different methods in the reference,the weights of these methods are compared.Using the CW value and each weight value in this study,the relative entropy values of weights are computed by the new weight evaluation approach.The results are 0.134,0.270,0.0694,and the results are consistent with the size relationship in the literature.The results are more prominent different,which proves that the pro‐posed approach is feasible.Since the multi-expert weight is composed of three subjective weights and CW is composed of four weights,two cases can be considered:three-weight and four-weight.The re‐sults are shown in Table 7.

When CW has only three weights,it is in the multi-expert weight mode,and the relative entropy values obtained by two weights show little difference(except the accuracy index).As the model coefficientsof the multi-expert weight are artificially set,the accuracy of the multi-expert weight is not high enough;thus,it is not convincing.When CW contains four weights,it is objective and more scientific.Compared with the mode of three weights,adding the fourth weight makes the relative entropy values of the accu‐racy index and stability index increase rapidly.The increase of device index and usability index is com‐mon.It shows that the introduction of OW has an impact on CW,which makes the distribution between CW and each weight worse.However,the objectivity of CW increases.

Table 7 Relative entropy values of weights of two groups

5 Summary

In this paper,a multiobjective optimization model of CW based on improved GT and an iMOEA/D-DE algorithm are proposed.The main contributions of this paper are as follows:

1.The multiobjective optimization model of CW based on improved GT is presented to overcome the drawback of poor objectivity of the multi-expert weight.The uncertainty of CW is also considered.

2.The iMOEA/D-DE algorithm is presented.First,the improved mutation operation is introduced to improve the convergence rate of the algorithm.Second,an adaptive strategy with self-learning abili‐ty is described to overcome the shortcomings thatFand CR in classical DE algorithms are constant val‐ues and that they cannot adapt to the multiobjective optimization model with nonlinearity and equality constraint.The adaptive strategy with self-learning ability depends on the changes of the fitness value within five generations.

3.A new weight evaluation approach based on relative entropy is presented to evaluate the rationality of the CW.

4.Experiments are carried out on test instances,on the CW model in the evaluation approach of the in‐tegrated navigation system and on the new weight evaluation approach.Results show that the proposed algorithm has excellent performance in certain as‐pects,as well as good distribution and convergence per‐formance.

In the future,the solution of multiobjective opti‐mization model of CW can be further improved.The main directions are proposed as follows:(1) The in-depth study on MOEA/D should be carried out to make the algorithm more suitable for the CW model,so as to be extended to multiobjective optimization problems with nonlinearity and equality constraint.(2)The new intelligent algorithms with better perfor‐mance should be used to solve the CW model.

Contributors

Mingtao DONG and Jianhua CHENG raised the research questions and the ideas to solve them.Mingtao DONG desi?gned the research.Jianhua CHENG processed the data.Mingtao DONG drafted the paper.Lin ZHAO helped organize the paper.Mingtao DONG and Jianhua CHENG revised and finalized the paper.

Compliance with ethics guidelines

Mingtao DONG,Jianhua CHENG,and Lin ZHAO declare that they have no conflict of interest.