ScienceDirect. Consistency simulation and optimization for HPIBM model in emergency decision making

Transcription

1 Available olie at ScieceDirect Procedia Computer Sciece 3 ( 4 ) d Iteratioal Coferece o Iformatio Techology ad Quatitative Maagemet, ITQM 4 Cosistecy simulatio ad optimizatio for HPIBM model i emergecy decisio makig Da Ergu a,b, *,Migsha Zhag a, Zhili Guo a, Xifeg Yag c a Southwest Uiversity for Natioalities, Chegdu, 64, Chia b School of maagemet ad ecoomics, Uiversity of Electroic Sciece ad Techology of Chia, 654, Chia c School of statistics, Southwester Uiversity of Fiace ad Ecoomics, Chegdu, 674 Chia Abstract Emergecies i ethic regios i Chia happeed icreasigly, ad emergecy decisio makig become more complicated tha that i other regios sice it ivolves religios, cultures ad atioal customs etc. The data cosistecy ad icompleteess issues are two of the hot research topics i a emergecy decisio matrix (EDM). The Hadamard product iduced bias matrix (HPIBM) model ca be used to idetify the most icosistet etry i a EDM. I this paper, we provide experimetal cosistecy simulatios for HPIBM by radomly geeratig decisio matrices with orders 3 to 7. I additio, we propose a ovel HPIBM for estimatig the missig judgmets i a EDM by costructig optimizatio problem while improvig the cosistecy ratio. A umerical example is used to illustrate the proposed method, ad the result shows that the proposed method ca fid the optimal solutios for missig values ad help emergecy maager make fast respose to the emergecy icidet eve there are icomplete decisio iformatio. 4 Published The Authors. by Elsevier Published B.V. by Ope Elsevier access B.V. uder CC BY-NC-ND licese. Selectio Selectio ad ad peer-review peer-review uder uder resposibility resposibility of of the the Orgaizig Orgaizig Committee Committee of ITQM of ITQM Keywords: Hadamard product iduced bias matrix; simulatio; emergecy decisio makig; icomplete matrix; missig comparisos. Itroductio Durig the last decades, emergecy happeed icreasigly aroud the world ad emergecy decisio makig has become oe of the hot research topics i the study of decisio makig -4. I recet years, emergecies i ethic * Correspodig author. Tel.: ; fax: address: erguda@63.com Published by Elsevier B.V. Ope access uder CC BY-NC-ND licese. Selectio ad peer-review uder resposibility of the Orgaizig Committee of ITQM 4. doi:.6/j.procs.4.5.3

2 Da Ergu et al. / Procedia Computer Sciece 3 ( 4 ) regios i Chia cotiue to occur, such as the 3.4 icidet i Tibet, 5. Wechua earthquake i year 8, the Urumqi "7.5" riots i year 9, Yushu earthquake i year, ad Ya a earthquake i year 3 etc. These icidets ot oly caused serious casualties ad ecoomic loss, but also affected the social stability ad ecoomic developmet i ethic miority areas 5. Differet from the other regios i Chia, ucovetioal icidets i the ethic regios have the followig features such as cocealmet, complexity ad situatioal characteristics, which make it more complicated ad difficult to cotrol 6-7. For example, a regular emergecy i ethic area could easily evolve to a ucovetioal icidet as log as it ivolves the atioal factors such as religio, culture ad customs differece etc. Therefore, it is very importat to study the emergecy decisio makig i ethic regios i Chia. I the process of decisio makig for icidet i the ethic regios, two of the well-kow multicriteria decisio makig methods, the Aalytic Hierarchy Process (AHP) 8 ad the Aalytic Network Process (ANP) 9, ca be used to idetify the key sceario factors ad make decisios -. However, as the three costraits i the emergecy decisio makig, i.e. time costrait, limited iformatio, ad decisio load costrait, decisio makers may make icosistet or icomplete judgmets -4. Therefore, the icosistecy ad icompleteess issues of a pairwise compariso i emergecy decisio makig become more serious tha that i regular decisio makig by AHP ad ANP 5-8. To idetify the most icosistet etry i the origial pairwise compariso matrix A, Kou et al. 9 proposed a Hadmard product iduced bias matrix (i short HPIBM hereiafter) model. The mathematical theorem has bee proved, ad some umerical examples have bee used to illustrate the proposed HPIBM model i Kou et al. 9. I this paper, we aim to achieve two objectives by hadlig the experimetal simulatio problem for cosistecy ad estimatio problem for missig judgmets. The first is to coduct experimetal simulatio o the HPIBM model for further validatig the effectiveess of the HPIBM whe it is used to improve the cosistecy. The simulatios are coducted by geeratig radomly matrices with orders 3 to 7. The secod goal of this paper is to propose a ovel HPIBM model for estimatig the missig comparisos i a emergecy decisio matrix. Simply speakig, the missig comparisos i a matrix are first completed by ukow variables, the apply the HPIBM model to costruct optimizatio problem to fid the optimal solutios. The rest of this paper is orgaized as follows. Next sectio first briefly describes the theorem of HPIBM, ad the the experimetal simulatio ad results are addressed. I Sectio 3, the HPIBM model is further exteded to estimate the missig comparisos i a icomplete emergecy decisio matrix. A umerical example is used to illustrate the proposed method i this sectio. We coclude this paper i Sectio 4.. Experimetal simulatio.. The theorem of Hadmard product iduced bias matrix (HPIBM) We first briefly preset the theorem of HPIBM 9, ad the coduct the cosistecy simulatios by radomly geeratig pairwise compariso matrices. Accordig to the theorem of HPIBM, the iduced bias matrix (IBM) C i eq. () with the equality symbol = will hold if the pairwise compariso matrix A is perfectly cosistet. If matrix A is just approximately cosistet, the the IBM C i eq.() with symbol will hold. Otherwise, there is at least oe elemet i each row (colum) of matrix C that deviates sigificatly from. U if aikakja T C AA A c ( aikakja ) () k U if aikakja where c aikakja ; U ; deotes the order of A; ad A T is the traspose of matrix A. k The symbol '' deotes Hadamard product, e.g. C A B idicatig c ab for all i ad j. The above theorem is proved mathematically ad illustrated by some umerical examples i Kou et al. 9, iterested readers are referred to Kou et al. 9 for details. I the followig, we coduct the cosistecy simulatios for

3 56 Da Ergu et al. / Procedia Computer Sciece 3 ( 4 ) radom emergecy decisio matrices whe the HPIBM is used to idetify the key sceario factors of emergecy icidet i ethic regios i Chia... The experimetal simulatios ad results To validate the correctiveess of the HPIBM model for improvig the cosistecy whe applyig it to emergecy decisio matrices, we coducted experimetal simulatios o icosistecy idetificatio ad adjustmet by radomly geeratig 5 set of reciprocal pairwise compariso matrix (PCM) with orders 3 to 7. The radom etries of the reciprocal PCM are subject to Saaty s 9-poit scale ad are radomly picked from the followig umbers, /9, /8, /7,,,, 3,, 9. After geeratig the radom PCM, the correspodig cosistecy ratio of each matrix is calculated, the apply the above HPIBM model to the matrices with CR>. i order to idetify ad adjust the most icosistet etry. The priciple of idetifyig ad adjustig the most icosistet etry is to idetify ad adjust the largest value i the HPIBM C (say c max ). If the cosistecy ratio of oe radom matrix caot be reduced ad lower tha the threshold. by adjustig the correspodig etry a i the radom matrix A, the select the secod, third largest etry i matrix C to improve its cosistecy ratio util it passes the cosistecy test. The results of experimetal simulatios are obtaied, as show i Table. We ca see from Table that the umbers of radom matrices for order 3 to order 7 with CR>. are 79387, 9677, 99744, ad respectively. Apply the HPIBM model to idetify ad adjust the most icosistet etries i these matrices, we foud that all the revised radom matrices passed the cosistecy test ad their cosistecy ratios have bee improved ad lower tha.. The correspodig simulatio for matrices with differet orders are plotted ad show i Figures -5. Table. Simulatio experimet for radomly geerated matrices with differet orders Matrix Order Number of simulatio Number of matrices with CR>. Failed matrices Passed Matrices CR>.. CR after adjustmet The umber of simulatio x 4 The umber of simulatio x 4 Fig. Experimetal simulatio experimet for 5 radomly geerated matrices with order 3

4 Da Ergu et al. / Procedia Computer Sciece 3 ( 4 ) CR>.. CR after adjustmet The umber of simulatio x 4 The umber of simulatio x 4 Fig. Experimetal simulatio experimet for 5 radomly geerated matrices with order 4 3 CR>. X: 9.499e+4 Y: CR after adjustmet The umber of simulatio x 4 The umber of simulatio x 4 Fig.3 Experimetal simulatio experimet for 5 radomly geerated matrices with order 5

5 56 Da Ergu et al. / Procedia Computer Sciece 3 ( 4 ) CR>. X: 6.33e+4 Y:.3.5 CR after adjustmet The umber of simulatio x 4 The umber of simulatio x 4 Fig.4 Experimetal simulatio experimet for 5 radomly geerated matrices with order 6.5 CR>..5 CR after adjustmet X: 8.94e+4 Y: The umber of simulatio x 4 The umber of simulatio x 4 Fig.5 Experimetal simulatio experimet for 5 radomly geerated matrices with order 7 3. HPIBM for estimatig missig comparisos 3.. The priciple of HPIBM for missig comparisos I the process of emergecy decisio makig for the emergecy icidets happeed i the ethic regios, decisio makers may provide a icomplete decisio makig matrix because of the limited decisio iformatio, time limitatio, religios ad culture coflicts etc. Therefore, the missig comparisos i a emergecy decisio matrix (EDM) should be estimated before makig a valid emergecy decisio. Based o the theorem of HPIBM model, we further exted it ito the case of estimatig the missig comparisos i a EDM. Theorem : The Hadarmad product iduced bias error matrix (HPIBEM) should be equal (or close) to a zero matrix if judgmet matrix A is perfectly (or approximately) cosistet, that is,

6 Da Ergu et al. / Procedia Computer Sciece 3 ( 4 ) if aik akja T AA A -U c ( aik akja ) () k if aik akja where A ad A T deote the revised complete matrix A with ukow variables x ad /x ; x ad /x ; etc, ad the correspodig traspose matrix respectively. Proof: For the case that judgmet matrix A is perfectly (or approximately) cosistet, Kou et al. 9 have proved that c aikakja. Obviously, we ca obtai that, k if aikakja aikakja. k if aikakja Accordig to the above HPIBM for missig comparisos, we ca estimate the missig values by costructig the optimizatio problem ad miimizig the error matrix ε, that is, Mi a, x x, x, x p, a aikakja (3) k s. t. / 9 x 9 where a ik, a kj, a may cotai ukow variables x, x etc. if the correspodig etry is missig. I practice, the least square method (LSM) ca be used to the above optimizatio problem, the we ca costruct the followig oe objective fuctio to fid the optimal solutios, a, x a, x Mi f aikakja j i j i k (4) s. t. / 9 x 9 To summarize, the processes of estimatig the missig compariso i a icomplete EDM by HPIBM model iclude: Step : Costruct the Quasi-complete matrix by pairs of ukow variables. Step : Establish optimizatio problem by HPIBM model. Step 3: Solve the optimal solutio of the optimizatio problem. Step 4: Test the cosistecy of the revised compariso matrix. I the followig sectio, a umerical example is used to illustrate the proposed method. 3.. Numerical example Assume there are five emergecy resposes alteratives, deoted as E, E, E3, E4 ad E5, ad a emergecy expert provided the followig emergecy decisio matrix A with maximum eigevalue 5.3 ad CR=.67, which is already less tha the threshold.. To illustrate the processes of estimatig the missig comparisos by HPIBM, suppose the followig five comparisos are missig, i.e., a 4, a 3, a 5, a 34, a 45, ad the icomplete EDM is writte as A( ), A , A..667 (5)

7 564 Da Ergu et al. / Procedia Computer Sciece 3 ( 4 ) Apply the proposed HPIBM model to this matrix ad follow the above four steps, we ca fid the optimal solutios. Details are show below. Step : Costruct the Quasi-complete matrix B by fillig i the missig comparisos with ukow variables x i (i=,..5) ad their reciprocals, we have, x x.3333 x3 B. / x x.667 (6) 4 / x 3 / x4 x5.474 / x3 6 / x5 Step : Establish optimizatio problem by HPIBM model. Mi s. t. f b, x b, x / 9 x i 9 j i 5 j i k b ik b kj b (7) Step 3: Fid the optimal solutios of the above optimizatio problem. For this optimizatio problem, we apply the fmico (algorithm: 'medium-scale: SQP, Quasi-Newto, liesearch') i Matlab software to estimate the optimal solutios. The results of optimizatio are show i Table, ad plotted i Figure 6. Figure 6(a) shows that the aggregatio values of objective fuctio f(x) reduced slightly after the th iteratio ( th dot i the x axis) ad stopped after 6 times of iteratios. More precisely, Table shows that aggregatio values of objective fuctio f(x) is less that after the th iteratio, ad become almost stable after the 9 th iteratios. The estimated optimal solutios of five ukow variables, a 4, a 3, a 5, a 34, a 45 are.859,.695,.333,.967, ad.9986 respectively. These optimal solutios are close to the correspodig origial values.87,.7778,.3333,.49 ad.88. Step 4: Test the revised complete matrix by the optimal solutios. We ca get that the λ max = 5.6, ad the CR=.36, idicatig the revised complete matrix passed the cosistecy test. 3 Curret Poit Curret poit Fuctio value (b) Number of variables: 5 Curret Fuctio Value: (a) Iteratio Fig. 6 Chages of objective fuctios ad the estimated optimal solutio

8 Da Ergu et al. / Procedia Computer Sciece 3 ( 4 ) Table. The aggregatio values of objective fuctios ad the optimal solutio Iter f(x) Iter f(x) Iter f(x) Iter f(x) Iter f(x) variables x i Estimated values x x x x x It ca be see from the experimetal results that the five assumed missig comparisos ca effectively be estimated by the ovel HPIBM model. I additio, the estimated values are very close to the origial values i matrix A. Whe replacig the missig values by these estimated values, the revised matrix successfully passed the cosistecy test, therefore, the ovel HPIBM for missig comparisos is effective ad efficiet. To make a compariso with the commoly used method i the Super Decisio makig software (i short SD), we calculated the priority vectors for the icomplete matrix by SD software, the applyig the estimatig formula x =ω i /ω j to estimate the missig comparisos. The compariso results are show i Table 3. Obviously, although the estimated values by both methods are close to each other, except x, the values of x, x 3, x 4 ad x 5 estimated by HPIBM method are closer to the origial values tha the values estimated by SD method. Therefore, the proposed method is accurate ad efficiet. Table 3. The compariso results of the estimated missig comparisos by HPIBM method ad SD method Ukow variables Priority vectors (ω i ) SD method HPIBM model Origial values x x x x x Coclusios The icosistecy ad icompleteess of a emergecy decisio matrix becomes more serious i ethics regios because of the atioal factors such as religio coflicts, culture differeces ad atioal customs etc. I this paper, the cosistecy simulatio is coducted to validate the capability of Hadamard product iduced bias matrix (HPIBM) model for improvig the cosistecy ratios. To simulate all possible cases of emergecy decisio matrices provided by emergecy expert, 5 umbers of reciprocal pairwise compariso matrices with orders 3 to 7 are radomly geerated, ad radom matrices with CR>. are used to test the proposed HPIBM model. The experimetal results show that all revised matrices with orders 3 to 7 passed the cosistecy test ad their cosistecy ratios have bee improved. I additio to experimetal simulatio, the HPIBM model is exteded to estimate the missig comparisos i a icomplete emergecy decisio matrix. Briefly speakig, the missig comparisos are filled i by ukow variables, the apply the HPIBM model to the quasi-complete matrix, fially costruct optimizatio problems by least square method to fid the optimal solutios. The proposed HPIBM is a effective way to deal with the icosistecy ad icompleteess issues i a emergecy decisio matrix. Although the experimetal simulatio ad umerical example for estimatig missig comparisos validated that the HPIBM is a effective ad accurate model for improvig cosistecy ratio ad estimatig missig comparisos i a EDM, the simulatio for radom matrices with higher orders ad the case study i real world emergecy decisio makig are our ext research directios.

9 566 Da Ergu et al. / Procedia Computer Sciece 3 ( 4 ) Ackowledgemets This research has bee partially supported by grats from the Natioal Natural Sciece Foudatio of Chia (#73736, #94,#78, ad #735), ad Academic Degree Programs Costructio at Southwest Uiversity for Natioalities (#4XWD-S, B34). Refereces. Cauhye, A. M., Nie, X., Pokharel, S. Optimizatio models i emergecy logistics: A literature review. Socio-Ecoomic Plaig Scieces, ; 46(), Jacobso, E.U., Argo, N.T., Iya, S. Priority Assigmet i Emergecy Respose, Operatios Research ; 6 (4): Ju, Y.B. Wag, A.H. Emergecy alterative evaluatio uder group decisio makers: A method of icorporatig DS/AHP with exteded TOPSIS. Expert Systems with Applicatios ; 39: Peg, Y., Kou,G., Wag, G., Wu, W., ad Shi, Y. Esemble of software defect predictors: a AHP-based evaluatio method, Iteratioal Joural of Iformatio Techology & Decisio Makig ; (): : Yu J., Liu, Y.K., Wag D.G., He, L.J. Research o Emergecy Maagemet of Ucovetioal Emergecies i Ethic Areas Based o Complex System, Chia Safety Sciece Joural ; (3): 7-75 (i Chiese). 6. Yag, A.H. Research o Icidets i Ethic Areas ad the Crisis Maagemet System, Joural of Public Maagemet 5; (3): 4-48 (i Chiese). 7. Xu J.H. A first exploratio o the fudametal theory of emergecies ivolvig i atioal elemets, Heilogag Natioal Series ; 6: 9-3 (i Chiese). 8. Saaty, T. L. How to Make a Decisio: The Aalytic Hierarchy Process. Iterfaces 994; 4: Saaty, T. L., 996. Decisio Makig with Depedece ad Feedback: The Aalytic Network Process. Pittsburgh, Pesylvaia: RWS Publicatios. ISBN Peg, Y., Kou, G., Wag, G., ad Shi, Y. FAMCDM: A Fusio Approach of MCDM Methods to Rak Multiclass Classificatio Algorithms, Omega ; 39(6): Kou, G., Lu, Y., Peg, Y., ad Shi, Y. Evaluatio of Classificatio Algorithms usig MCDM ad Rak Correlatio, Iteratioal Joural of Iformatio Techology & Decisio Makig ; (): Yu, L.A, Lai, K.K. A distace-based group decisio-makig methodology for multi-perso multi-criteria emergecy decisio support, Decisio Support Systems ; 5: Ergu, D., Kou, G., Peg, Y., Li, F., Shi, Y. Data cosistecy i emergecy maagemet. It J Comput Commu Cotrol ; 7(3), Peg, Y., Kou,G., Shi, Y., ad Che, Z. A Descriptive Framework for the Field of Data Miig ad Kowledge Discovery, Iteratioal Joural of Iformatio Techology & Decisio Makig 8;7(4): Ergu, D., Kou, G., Peg, Y., Shi, Y. A Simple Method to Improve the Cosistecy Ratio of the Pair-wise Compariso Matrix i ANP, Europea Joural of Operatioal Research ; 3(): Altuzarra, A., Moreo-Jiméez, J. M., Salvador, M. Cosesus Buildig i AHP-Group Decisio Makig: A Bayesia Approach. Operatios Research ; 58 (6): Gomez-Ruiz, J. A., Karaik, M., Peláez, J.I. Estimatio of missig judgmets i AHP pairwise matrices usig a eural etwork-based model. Applied Mathematics ad Computatio ; 6: Kou, G., Li, C. A cosie maximizatio method for the priority vector derivatio i AHP, Europea Joural of Operatioal Research 4; 35: Kou, G., D. Ergu, Shag, J. Ehacig Data Cosistecy i Decisio Matrix: Adaptig Hadamard Model to Mitigate Judgmet Cotradictio. Europea Joural of Operatioal Research 3;