Research Article Efficiency Bounds for Two-Stage Production Systems

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1 Matheatical Proble in Engineering Volue 2018, Article I , 9 page Reearch Article Efficiency Bound for Two-Stage Production Syte Xiao Shi School of Finance, Shandong Univerity of Finance and Econoic, Jinan, Shandong Province , China Correpondence hould be addreed to Xiao Shi; hixiao@dufe.edu.cn Received 17 January 2018; Revied 2 July 2018; Accepted 12 July 2018; Publihed 26 July 2018 Acadeic Editor: Vladiir Turetky Copyright 2018 Xiao Shi. Thi i an open acce article ditributed under the Creative Coon Attribution Licene, which perit unretricted ue, ditribution, and reproduction in any ediu, provided the original work i properly cited. Traditional data envelopent analyi (EA) odel find the ot deirable weight for each deciion-aking unit (MU) in order to etiate the highet efficiency core a poible. Thee efficiency core are then ued for ranking the MU. The ain drawback i that the efficiency core baed on weight obtained fro the tandard EA odel ignore other feaible weight; thi i due to the fact that EA ay have ultiple olution for each MU. To overcoe thi proble, Salo and Punkka (2011) deeed each MU a a Black Box and developed odel to obtain the efficiency bound for each MU over et of all it feaible weight. In any real world application, there are MU that have a two-tage production yte. In thi paper, we extend the Salo and Punkka (2011) odel to a ore coon and practical cae conidering the two-tage production tructure. The propoed approach calculate each MU efficiency bound for the overall yte a well a efficiency bound for each ubyte/ubtage. An application for nonlife inurance copanie ha been dicued to illutrate the applicability of the propoed approach and how the uefulne of thi ethod. 1. Introduction ata Envelopent Analyi (EA), firt developed by Charne et al. [1], ha been proven a an effective tool for perforance evaluation and bencharking. Thi technique ake no auption on the production function and ipoe no ubjective weight on ultiple input and ultiple output. EA ha been widely applied in any area [2]. The EA technique allow a MU to chooe the ot favorable weight to achieve the bet poible relative efficiency. However, the tandard efficiency core do not conider all the poible weight a they only conider the weight ot favorable to each MU. The ain iue that ha been ignored in the pat literature i that the efficiency core of a MU relative to other MU can change over different weight when applying the EA odel. Hence, it i iportant to conider all poible weight to evaluate each MU. To overcoe thi proble, Salo and Punkka [3] have propoed a procedure to obtain the efficiency bound by taking into account all poible weight (ee alo [4]). That i to ay, the efficiency bound how how the MU efficiency ratio relate to each other for all feaible weight, rather than for thoe weight only for which the data envelopent analyi (EA) efficiency core of oe MU i axiized. They have introduced an efficiency bound for all poible efficiency core that i deterined by the lower and upper bound of the efficiency core. The efficiency bound how how uch ore efficient a given MU can be relative to oe other MU or a ubet of other MU. For thi purpoe, Green et al. [5] developed a new odel by putting the CCR odel into a ixed-binary linear prograing fraework to obtain the efficiency bound in data envelopent analyi. Entani and Tanaka [6] propoed the interval EA odel to obtain an efficiency interval coniting of evaluation fro both the optiitic and peiitic viewpoint. Wang and Yang [7] propoed a pair of bounded EA odel to eaure the overall perforance of a group of deciion-aking unit (MU), which were characterized by interval efficiencie. To overcoe the proble of thee odel incapable of deterining an efficiency interval for any MU when there i a zero value for each output, Azizi and Wang [8] propoed a pair of iproved bounded EA odel to overcoe the drawback. All thee ethod for obtaining the efficiency bound treated each MU a a Black Box. Thu, they ignored the internal tructure of the production yte. However, a dicued in any EA tudie, in any real application, MU have a two-tage tructure, i.e., output

2 2 Matheatical Proble in Engineering fro the firt tage becoe the input to the econd tage. Output fro the firt tage are referred to a interediate eaure. Seiford and Zhu [9] ue the tandard EA approach to eaure the profitability and arketability of US coercial bank which doe not addre potential conflict between the two tage ariing fro the interediate eaure. For exaple, the econd tage ay have to reduce it input (interediate eaure) in order to achieve an efficient tatu, which iply a reduction in the firt tage output. In a urvey by Cook et al. [10], they pointed out that the approache of odeling MU with a twotage production proce can be categorized a four type, i.e., tandard EA ethodology, efficiency decopoition ethodology, network EA, and gae-theoretic approache. The tandard EA ethodology iply ue the tandard EA odel, i.e., two eparate EA odel to calculate the efficiencie of two tage (e.g., Seiford and Zhu [9]; Zhu [11]; and Sexton and Lewi [12]); the efficiency decopoition ethodology i that given the efficiency core of tage 1 and tage 2, the overall efficiency could be defined a the product or the arithetic ean of two ubtage efficiencie (e.g., Kao and Hwang [13]; Chen et al. [14], and Chen et al. [15]); the network EA approach extend the two-tage proce to ore general ituation (e.g., Tone and Tutui [16]; Tone and Tutui [17]; Izadikhah et al. [18]); gae-theoretic approache introduce gae theory to the efficiency evaluation of twotage tructure (e.g., Liang et al. [19]; Zha and Liang [20]; Li et al. [21]; Guo and Zhu [22]; and Izadikhah et al. [18]). Except for the tandard EA approach, all other approache attept to correct for the above-referenced conflict iue. And, twotage EA ha been extenively applied to any area, uch a hotel ([23, 24]; Huang et al. [25]), R& departent (Li et al. [21]; Liu and Lu, [26]), inforation technology (Shao and Lin [27]; Chen and Zhu [28]; and Kao and Hwang [29]), inurance copanie (Yang [30]; Kao and Hwang [13]), indutry (Wu et al. [31]; Chen et al. [32]; and Li et al. [33]), and bank (Paradi et al. [34]; Huang et al. [23, 24]; Wang et al. [35]; and Zhu et al. [36]). In thi paper, we develop a ethod to obtain the efficiency bound for the claic two-tage production yte adicuedbyseifordandzhu[9],chenandzhu[28], and Kao and Hwang [13]. That i, the firt ubyte ue input to produce output that then becoe the input to the econd ubyte to produce the final output. The propoed odel calculate each MU efficiency bound for the overall yte a well a two ubyte. Unlike conventional efficiency core, the reult how how the MU efficiency ratio for the overall yte and two ubyte relate to each other for all feaible weight. We believe that thi proce provide ore accurate inforation for deciion aker by identifying the bet (and/or wort) MU in the overall yte and both ubyte over all feaible weight. Beide, the propoed approach provide inforation regarding the enitivity of the MU efficiency bound for the overall yte and both two ubyte over et of all feaible weight. The reainder of thi paper i organized a follow. In the next ection, the procedure of efficiency bound by Salo and Punkka [3] ha been reviewed briefly. Then, in Section 3, a ethod i developed to obtain the efficiency bound conidering the two-tage production yte. Thi i followed by illutration exaple in Section 4. An application i alo given in thi ection to how the uefulne of the propoed procedure. Finally, concluion and direction for future reearch are given in Section Efficiency Bound by Salo and Punkka [3] Aue that there are n MU denoted a MU j (j = 1,...,n).EachMUueinputx ij (,...,)to produce output y rj (r = 1,...,). Baed on the definition of Charne et al. [1], the efficiency of MU k i calculated by the CCR ultiplier for a follow: E CCR k (u, V) = ax.t. u r y rj u r 0 V i 0 u ry rk V ix ik (1) r i V i x ij 0 j (2) The odel could be tranfored into linear prograing odel by ue of the Charne-Cooper tranforation [37]. u r (r = 1,...,) and V i (i = 1,...,) are the optial olution of odel (2) and the aociated optial output and input weight. MU k i tered efficient if and only if the optial objective i equal to one and the optial weight vector are all larger than zero; i.e., E CCR k (u, V ) = 1 and u r > 0,V i > 0. For any feaible weight u r (r = 1,...,) and V i (i = 1,...,), Salo and Punkka [3] defined efficiency doinance between MU baed on the efficiency core of odel (2). efinition 1. MU k doinate MU l (denoted by MU k MU l )ifandonlyif E k (u, V) E l (u, V) (u, V) (S u,s V ) E k (u, V) >E l (u, V) for oe (u, V) (S u,s V ) Thu, if MU k MU l, the efficiency ratio of MU k i atleatahighathatofmu l for all feaible weight, and oreover, there exit oe weight for which it efficiency i trictly higher. The doinance relation in efinition 1 could be calculated baed on the following pairwie efficiency ratio: (3) (4) k,l (u, V) = E k (u, V) E l (u, V). (5) However, the relative efficiency ratio (5) i nonlinear in weight (u, V). Salo and Punkka [3] propoed the following odel to axiize and iniize the ratio through linear prograing.

3 Matheatical Proble in Engineering 3 Theore 2. The optiu of the axiization (iniization proble) ax u,v.t. (in u,v ) u r y rk u r y rl V i x il u r, V i 0, r,i i the axiu of (iniu) k,l (u, V) for all weight cenario, i.e., k,l and k,l. The optiization proble in odel (6) obtain the upper and lower bound on how efficient MU k can be relative to MU l acro feaible weight. It i worth noting that if the benchark et L contain all MU, then k,l (or k,l ) i equal to the CCR-EA core. If MU k i not contained in the benchark et L, k,l (or k,l ) i the uper efficiency of MU k relative to thi et of MU (ee, e.g., [38]). For exaple, if k,l = 1.2, the efficiency core of MU k i at leat 20% higher than that of MU l.and,if k,l =1.5,the efficiency core of MU k can be at ot 50% higher than that of MU l.iftheiniu k,l i greater than one, MU k doinate MU l.iftheiniu k,l i le than one, the doinance doe not hold. If the iniu i equal to one, we could judge the doinance relation by further axiizing the linear progra of odel (6). If the reulting axiu k,l i greater than one, the doinance hold, but if not, then MU k and MU l have the ae efficiency core (2) for all feaible weight. Baed on odel (6), the lower efficiency bound of MU k, which i the efficiency of MU k relative to the ot efficient MU in the benchark group for different input/output weight. Thu, the lower bound of MU k efficiency core denoted by k,l i in l L k,l, i.e., k,l = in l L k,l. The following propoition how how to find the upper bound of efficiency core, denoted a k,l,whichirelative to the ot efficient MU in the benchark group for MU k. Theore 3. k,l i the optiu of the axiization proble ax u,v.t. u r y rk u r y rl V i x il, l L u r, V i 0, r,i. (6) (7) The optial value of odel (7) i the upper bound of MU k efficiency core over et of all feaible weight. The upper bound of MU k efficiency core defined a the axiu of value of how efficient MU k i relative to the ot efficient MU in the benchark group for different input/output weight. Baed on odel (6) and (7), the efficiency bound [ k,l, k,l ] of MU k can be coputed. Thee odel treat the production yte a Black Box. The approach can be generalized to yte copoed of two ubyte connected in erie. In the next ection, we will dicu how to calculate the efficiency bound for each MU with a two-tage production yte. 3. Efficiency Bound for Two-Stage Production Syte Suppoe the operation of a MU can be divided into two ubyte or procee, a depicted in Figure 1. For MU k, ubyte 1 applie input x ik (i = 1,..., ) to produce the interediate product z dk (d = 1,..., ). All thee interediate product are then ued by ubyte 2 to produce the final output y rk (r = 1,..., ). Baedonthe definition of Kao and Huang [13], MU k efficiency core for the overall yte and two ubyte are defined a E k = r u ry rk i V i x ik (8) E 1 k = d w1 d z dk i V i x ik (9) E 2 k = r u ry rk d w 2 d z dk (10) where u r (r = 1,..., ) and V i (i = 1,..., ) are the output weight and input weight, repectively. Accordingly, w 1 d (d = 1,..., ) and w 2 d (d = 1,..., ) are the weight attached to the interediate eaure for ubyte 1 and ubyte 2, repectively. Siilar to Kao and Hwang [13] and Liang et al. [19], we aue that the weight attached to the interediate output in both ubyte 1 and ubyte 2 are the ae, i.e., w 1 d =w2 d. Thi auption repreent the erial relationhip between the two ubyte [14]. If we olve the two-tage EA without thi auption, then our ethod i identical to independently eploying the odel for each ubyte. Therefore, thi paper aue w 1 d =w2 d =w d Efficiency Bound of a MU for the Overall Syte. A dicued in Section 2, chooing different weight ay lead to different efficiency core for a Black Box MU. Siilarly, chooing different weight ay reult in different efficiency core for a MU with a two-tage production yte. Propoition 4. The optiu of the axiization (or iniization) proble

4 4 Matheatical Proble in Engineering x ij (i= 1,...,) z dj (d= 1,...,) y rj (r= 1,...,) Sub-yte 1 Sub-yte 2 Figure 1: Two-tage production yte. ax u,v,w (in) u,v,w u r y rk in V,w w d z dk.t. u r y rl = w d z dl.t. w d z dj = V i x il (13) w d z dl = V i x il (11) u r, V i,w d 0, r,i,d w d, V i 0, d,i i the iniu of 1 k,l (w, V) for ubyte 1 for all weight cenario. i the axiu (or iniu) of k,l (u, w, V) for all weight cenario, i.e., k,l (u, w, V) (or k,l (u, w, V)).Theproofofthi propoition i given in the Appendix. In the following odel, the optiu of the axiization proble, i.e., k,l, i the upper efficiency bound of MU k conidering each MU two-tage production tructure. Propoition 5. k,l i the optiu of the axiization proble ax u,v.t. u r y rk u r y rl w d z dj w d z dl, V i x il, l L l L u r, V i 0, r,i. (12) By olving odel (12), the upper efficiency bound k,l could be obtained. The proof of thi propoition i given in the Appendix Efficiency Bound of a MU for Both Subyte. In thi ection, we dicu the efficiency bound of MU for two ubyte. Propoition 6. The optiu of the axiization (iniization proble) Baed on odel (13), the lower bound of MU k efficiency core for ubyte 1, which i the efficiency of MU k relative to the ot efficient MU in the benchark group for different input/output weight. Thu, the lower bound of MU k efficiency core for ubyte 1 denoted by 1 k,l i in l L 1 k,l, i.e., 1 = in k,l l L 1 k,l.theproofofthi propoition i iilar to the proof of Theore 3 in Salo and Punkka [3]. In the following odel, the optiu of the axiization proble, i.e., 1 k,l, i the upper bound of MU k efficiency core for ubyte one. Propoition 7. 1 k,l i the optiu of the axiization proble ax u,v.t. w d z dk w d z dj V i x il, l L w d, V i 0, d,i. (14) The optial value of odel (14) ay be greater than 1 or le than one or equal to 1. If the optial value i greater than 1, then MU k doinate MU l. The proof of thi propoition i iilar to the proof of Theore 4 in Salo and Punkka [3]. Siilarly, the lower and upper bound of MU k efficiency core for ubyte 2 are calculated by the following two linear progra.

5 Matheatical Proble in Engineering 5 Table 1: Ranking interval for each nonlife inurance copany. MU NO. Our approach Salo & Punkka [3] odel k,l (u, w, V) k,l (u, w, V) k,l (u, V) k,l (u, V) Taiwan Fire Chung Kuo Tai Ping China Mariner Fubon Zurich Taian Ming Tai Central The Firt Kuo Hua Union Shingkong South China Cathay Century Allianz Preident Newa AIU North Aerica Federal Royal&Sun Alliance Aia AXA Mitui Suitoo Propoition 8. The optiu of the iniization proble in u,w.t. u r y rk u r y rl = w d z dk =1 w d z dl u r, V i,w d 0, r,i,d i the iniu of 2 k,l (u, V) for all weight cenario. (15) Baed on odel (15), the lower bound of MU k efficiency core for ubyte 2, which i the efficiency of MU k relative to the ot efficient MU in the benchark group for different input/output weight. Thu, the lower bound of MU k efficiency core for ubyte 2 denoted by 2 i in k,l l L 2 k,l, i.e., 2 = in k,l l L 2 k,l. The proof of thi propoition i iilar to the proof of Theore 3 in Salo and Punkka [3]. Propoition 9. 2 i the optiu of the axiization k,l proble ax u,v.t. u r y rk u r y rl w d z dk =1 w d z dl, l L u r, V i 0, r,i. (16) Thu, the efficiency bound 2 k,l, 2 k,l of MU k for ubyte2couldbeobtainedbyodel(15)andodel(16). 4. Epirical Illutration To illutrate the propoed approach of efficiency bound for two-tage production yte, we ue the following exaple. In thi ection, we take the data et of 24 nonlife inurance copanie fro [13]. Thee nonlife inurance copanie whole production yte ha a typical two-tage tructure. The production yte i divided into two ubyte: preiu acquiition and profit generation. Thee copanie are evaluated by uing two input, two interediate, and two output. Table 1 report the efficiency bound of each MU baed on our approach and Salo and Punkka [3]

6 6 Matheatical Proble in Engineering Figure 2: Efficiency bound for each nonlife inurance copany conidering and without conidering the two-tage production yte. approach. Colun 5 and colun 6 in Table 1 report the MU efficiency bound treating the production yte a a Black Box, which are repreented graphically blue in Figure 2. Colun 3 and colun 4 in Table 1 report the MU efficiency bound when conidering the inner production tructure, which are repreented graphically red in Figure 2. Fro Figure 2, we can find that if we do not conider the two-tage production tructure, the bet MU are Chung Kuo (MU 2), Fubon (MU 5), Union (MU 12), and Aia (MU 22) a they have the bet efficiency core of 1. Though thee MU ay have the bet efficiency core of 1, they have wide efficiency bound. Aong thee five MU, Aia (MU 22) ha the bet perforance a it ha the narrowet efficiency bound a well a the bet efficiency core of 1. The wort MU are Tai Ping (MU 3), China Mariner (MU 4), Kuo Hua (MU 11), AXA (MU 23), and Mitui Suitoo (MU 24) a their lower efficiency core are le than 0.1. Aong thee five MU, the upper efficiency bound of Mitui Suitoo (MU 24) i , which i aller than that of other four MU. Thu, Mitui Suitoo (MU 24) i the wort MU. The bet perforer (or wort perforer) baed on our approachaynotbetheaeathatofsaloandpunkka [3] approach. The red bar char in Figure 2 report the MU efficiency bound when the two-tage tructure i conidered. Figure 2 how that Taiwan Fire (MU 1) ay be the bet NU a it upper efficiency ha the larget efficiency core of AXA (MU 23) ay be the wort MU a it lower efficiency ha the leat efficiency core of Beide, we can copare oe MU over et of all feaible weight. For exaple, Chung Kuo (MU 2) ha a bet efficiency core of and a wort efficiency core of , while Mitui Suitoo (MU 24) ha a bet efficiency core of and a wort efficiency core of That i, 24,L < 1,L and 24,L < 2,L < 2,L. Hence, Chung Kuo (MU 2) alway perfor better than Mitui Suitoo (MU 24) regardle of the choice of the weight. When the two-tage tructure i conidered, the efficiency bound ay be narrower baed on our approach than that of Salo and Punkka [3] approach. For exaple, a hown in Colun 3 and 4 in Table 1, it could be found that Taian (MU 7) ha the bet efficiency core of and the wort efficiency of However, a hown in Colun 5 and 6 in Table 1, it ha the bet efficiency core of and the wort efficiency core of over all feaible weight. Thu, the efficiency bound baed on Salo and Punkka [3] approach are wider. A for two ubyte, Table 2 how the ubyte efficiency bound of each inurance copany. The third and fourth colun in Table 2 are the lower and upper efficiency core for ubyte 1. The fifth and ixth colun in Table 2 are the lower and upper efficiency core for ubyte 2. It how that Central (MU 9), Union (MU 12), Cathay Century (MU 15), North Aerica (MU 19), and Mitui Suitoo (MU 24) are the bet perforer in ubyte 1. A Union (MU 12) ha the narrowet efficiency bound, it i the bet MU for ubyte 1. The ource of inefficiency could alo be identified; for exaple, Fubon (MU 5) ha the efficiency bound of [0.6035, ]. It can be een that it perfor well in ubyte 2 a the efficiency bound of ubyte 2 i [1, 1], while it doe not perfor well in ubyte 1 a the efficiency bound of ubyte 1 i [0.4468, ]. Therefore, the reaon why Fubon (MU 5) overall efficiency i o low i it bad perforance in ubyte 1. A for the ubyte 2, Tai Ping (MU 3), Fubon (MU 5),Newa(MU17),andAia(MU22)arethebetMU a their bet efficiency core are all 1. AXA (MU 23) and Mitui Suitoo (MU 24) ay be the wort MU a it wort efficiency core i the allet. 5. Concluion and irection for Future Reearch In previou EA literature, each MU i evaluated by uing the ot favorable weight. However, it ignore other feaible weight. To overcoe thi proble, Salo and Punkka [3] deeed each MU a a Black Box and developed a erie of odel to obtain the efficiency bound over et of all feaible weight. In thi paper, we expand their ethod by conidering the internal tructure of the MU. We extend their ethod to copute efficiency bound for a two-tage production yte and illutrate the ethod by reviiting reported EA tudie. Thu, the Black Box i opened, and ore accurate inforation on the efficiency bound for the overall yte and both ubyte i provided to the deciion aker. Unlike conventional efficiency core, the reult how how the MU efficiency ratio for the overall yte and two ubyte relate to each other for all feaible weight. The efficiency eaure ued in thi paper i radial; oe nonradial eaure have alo been propoed in the literature, uch a the lack-baed eaure [16, 17]. Obtaining the efficiency bound for two-tage production yte baed on nonradial EA i another intereting avenue to explore in the future. Appendix Proof of Propoition 4. Chooe the weight (u,w, V ) atifying k,l (u,w, V ) k,l (u, w, V), (u, w, V) are the

7 Matheatical Proble in Engineering 7 Table 2: Efficiency bound for two ubyte. MU NO. Efficiency bound for ub-yte 1 Efficiency bound for ub-yte k,l k,l 2 2 k,l k,l Taiwan Fire Chung Kuo Tai Ping China Mariner Fubon Zurich Taian Ming Tai Central The Firt Kuo Hua Union Shingkong South China Cathay Century Allianz Preident Newa AIU North Aerica Federal Royal&Sun Alliance Aia AXA Mitui Suitoo arbitrary weight. We define V i = V i / V i x ik. So, V i x ik = 1. We then define w d = w d V i x il/ w d z dl. Thu, w d z dl = V i x il. efine u r = u r w d z dl/ u r y rl;thu, u r y rl = w d z dl.the weight (u,w, V ) atify five contraint in odel (11). And by Lea 1 in Salo and Punkka [3], k,l (u,w, V ) = k,l (u,w, V )= u r y rk. So the axiu of odel (11) over the five contraint in odel (11) i at leat a high a k,l (u,w, V ). Aue the axiu of odel (6) i attained at (u 0,w 0, V 0 ).So,wehave k,l (u 0,w 0, V 0 )= E k (u 0,w 0, V 0 ) E l (u 0,w 0, V 0 ) = u k ry rk w dz dk k k w dz dk w dz dl V ix ik V ix il u ry rl k w dz dl = u 0 r y rk. (A.1) The weight (u 0,w 0, V 0 ) atify the five contraint in odel (11). Thu, axiu of k,l (u, w, V) over all the feaible weight would be larger or equal to the olution of the axiization proble in odel (11). The proof of the iniization proble could be hown in the analogou way. Proof of Propoition 5. k,l (u, w, V) = E k (u, w, V)/ ax l L E l (u, w, V). Let the axiu of k,l (u, w, V) be ζ. Thu, the optiu i attained at (u,w, V ). There then exit oe l L uch that E l (u,w, V ) E l (u,w, V ) l L.ChooeV i = V i / V i x ik, othat V i x ik = 1. Alo, chooe a contant c w > 0 o that w d z dl = V i x il for w = c w w. Alo, chooe a contant c u >0o that u r y rl = w d z dl for u = c u u.foranyl L,wehave 1 l,l (u, V )= l,l (u, V )= E l (u, V ) E l (u, V ) = u r y rl w d z. w d z dl dl V i x il. V i x il w d z = u r y rl dl w d z dl. w d z dl u r y rl. w d z dl V i x. il (A.2) So, the contraint u ry rl w dz dl,l L and w dz dj V ix il,l Lare atified by (u,w, V ).By contruction, ζ = ax u,w,v k,l (u, w, V) = k,l (u, V ) = u r y rk, which how the axiu of odel (12) i at leatahighathatof k,l (u, w, V).

8 8 Matheatical Proble in Engineering Converely, aue that the axiu of odel (12), ζ,iattainedat(u,w, V ),andchooel L o that the contraint in odel (12) i binding ( uch that l exit, for otherwie u could be increaed to iprove the value of the objective function, which would be in violation of the optiality auption). Now, ax u,w,v k,l (u, w, V) E k (u,w, V )/E l (u,w, V ) = ζ, o that the axiu of k,l (u, w, V)utbeatleatahighathatofodel(12). ata Availability The data i fro the data et of 24 nonlife inurance copanie fro [13] or fro the correponding author upon requet. Conflict of Interet There are no conflict of interet related to thi paper. Acknowledgent Thi reearch wa upported by Fund of National Natural Science Foundation of China (no , , and ).The fund were involved in the anucript writing, editing, approval, or deciion to publih. Reference [1]A.Charne,W.W.Cooper,andE.Rhode, Meauringthe efficiency of deciion aking unit, European Operational Reearch,vol.2,no.6,pp ,1978. [2] A. Erouznejad and G.-L. Yang, A urvey and analyi of the firt 40 year of cholarly literature in EA: , Socio- Econoic Planning Science,vol.61,pp.4 8,2018. [3] A. Salo and A. Punkka, Ranking interval and doinance relation for ratio-baed efficiency analyi, Manageent Science, vol.57,no.1,pp ,2011. [4] F. Yang, S. Ang, Q. Xia, and C. Yang, Ranking MU by uing interval EA cro efficiency atrix with acceptability analyi, European Operational Reearch, vol.223,no.2,pp , [5]R.H.Green,J.R.oyle,andW..Cook, Efficiencybound in data envelopent analyi, European Operational Reearch,vol.89,no.3,pp ,1996. [6] T. Entani and H. Tanaka, Iproveent of efficiency interval baed on EA by adjuting input and output, European Operational Reearch, vol.172,no.3,pp , [7] Y.-M. Wang and J.-B. Yang, Meauring the perforance of deciion aking unit uing interval efficiencie, Coputational and Applied Matheatic,vol.198,no.1,pp , [8] H. Azizi and Y.-M. Wang, Iproved EA odel for eauring interval efficiencie of deciion-aking unit, Meaureent,vol.46,no.3,pp ,2013. [9] L. M. Seiford and J. Zhu, Profitability and arketability of the top 55 U.S. coercial bank, Manageent Science,vol.45,no. 9, pp , [10] W.. Cook, L. Liang, and J. Zhu, Meauring perforance of two-tage network tructure by EA: a review and future perpective, Oega,vol.38,no.6,pp ,2010. [11] J. Zhu, Multi-factor perforance eaure odel with an application to Fortune 500 copanie, European Operational Reearch,vol.123,no.1,pp ,2000. [12] T. R. Sexton and H. F. Lewi, Two-tage EA: an application to ajor league baeball, Productivity Analyi, vol.19, no.2-3,pp ,2003. [13] C. Kao and S.-N. Hwang, Efficiency decopoition in twotage data envelopent analyi: an application to non-life inurance copanie in Taiwan, European Operational Reearch,vol.185,no.1,pp ,2008. [14]Y.Chen,W..Cook,N.Li,andJ.Zhu, Additiveefficiency decopoitionintwo-tageea, European Operational Reearch, vol. 196, no. 3, pp , [15] Y.Chen,L.Liang,andJ.Zhu, Equivalenceintwo-tageEA approache, European Operational Reearch,vol. 193, no. 2, pp , [16] K. Tone and M. Tutui, Network EA: a lack-baed eaure approach, European Operational Reearch, vol. 197, no. 1, pp , [17] K. Tone and M. Tutui, ynaic EA: a lack-baed eaure approach, Oega, vol. 38, no. 3-4,pp ,2010. [18]M.Izadikhah,M.Tavana,..Caprio,andF.J.Santo- Arteaga, A novel two-tage EA production odel with freely ditributed initial input and hared interediate output, Expert Syte with Application,vol.99,pp ,2018. [19] L. Liang, W.. Cook, and J. Zhu, EA odel for twotage procee: Gae approach and efficiency decopoition, Naval Reearch Logitic (NRL),vol.55,no.7,pp ,2008. [20] Y. Zha and L. Liang, Two-tage cooperation odel with input freely ditributed aong the tage, European Operational Reearch, vol. 205, no. 2, pp , [21] Y. Li, Y. Chen, L. Liang, and J. Xie, EA odel for extended two-tage network tructure, Oega, vol. 40, no. 5, pp , [22] C. Guo and J. Zhu, Non-cooperative two-tage network EA odel: Linear v. paraetric linear, European Operational Reearch,vol.258,no.1,pp ,2017. [23] C.-W.Huang,H.F.Ho,andY.-H.Chiu, Meaureentoftourit hotel productive efficiency, occupancy, and catering ervice effectivene uing a odified two-tage EA odel in Taiwan, Oega,vol.48,pp.49 59,2014. [24] J. H. Huang, J. J. Chen, and Z. J. Yin, A network EA odel with uper efficiency and undeirable output: an application to bank efficiency in China, Matheatical Proble in Engineering,vol.2014,pp.1 14,2014. [25] C.-W. Huang, Y.-H. Chiu, C.-H. Tu, Z.-Y. Luo, and Z. Wang, Uing the nonhoogeneou frontier two-tage EA odel to ae the efficiencie of expene utilization and operation of the Taiwanee hotel indutry, International Tranaction in Operational Reearch,vol.23,no.6,pp ,2016. [26] J. S. Liu and W. M. Lu, EA and ranking with the networkbaed approach: a cae of R& perforance, Oega, vol.38, pp , [27] B. B. M. Shao and W. T. Lin, Technical efficiency analyi of inforation technology invetent: A two-tage epirical invetigation, Inforation and Manageent, vol.39,no.5,pp , 2002.

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