Available nline a hp://jnrm.srbiau.ac.ir Vl.1, N.2, Summer 2015 Jurnal f New Researches in Mahemaics Science and Research Branch (IAU Prduciviy changes f unis: A direcinal measure f cs Malmquis index G. Thidi a *, S. Thidnia b (a,b Deparmen f Mahemaics, Cenral Tehran Branch, Islamic Azad Universiy, Tehran, Iran Received Spring 2015, Acceped Summer 2015 Absrac This sudy examines he prduciviy changes f decisin making unis in siuain where inpu price vecrs are varying beween hem and inpus are heergeneus; ha is a nncmpeiive marke. We presen a direcinal measure f cs Malmquis prduciviy index where incrpraes he decisin maker's preference ver prduciviy change ver ime. A simple numerical example is designed illusrae he new measure f he cs Malmquis index. Keywrds: Daa envelpmen analysis; direcinal measure; prfi efficiency; prduciviy change; cs Malmquis index *Crrespnding auhr: Email: ghahidi@yah.cm (gh_hidi@iaucb.ac.ir
G. Thidi, e al /JNRM Vl.1, N.2, Summer 2015 56 1. Inrducin The prduciviy change ver ime is an impran subjec fr decisin making unis (DMUs. Daa envelpmen analysis (DEA can be applied as a nn-parameric apprach in sudying he prduciviy change and is decmpsiin. DEA fr he firs ime was develped by Charnes e al. (1978 and has been used s far in many lieraures ha fcus n he efficiency and prduciviy f DMUs. Fr example, Berger (2007, Sahye (2003, Cli e al. (2011, Bruni e al. (2011. The mehd is als applied cmpue he Malmquis index ha is used fr evaluaing he prduciviy changes f DMUs ver ime. Caves e al. (1982 defined he Malmquis index based n efficiency scre fr he firs ime and hen Färe e al. (1989 decmpsed he index in efficiency and echnical change. Malmquis index is exended in differen ways. Fr insance, Chen (2003 presened he nn-radial Malmquis index and develped a nnradial DEA mdel fr cmpuing i. Arabi e al. (2015 as well as used a slacks-based measure (SBM mdel cmpue he prduciviy change f DMUs in he presence f undesirable upus. Maniadakis and Thanassulis (2004 exended he index he cs Malmquis (CM index fr he case where he prices f he inpus are knwn. They used he Farrell cs efficiency in definiin f he index. See als Prela and Thanassulis (2010, Thidi e al. (2010 fr her applicains f Malmquis index. The cs efficiency (CE measure used by Maniadakis and Thanassulis (2004 in definiin f CM index can be applied when inpus are hmgenus and he prices are exgenusly fixed. Thus, he CE measure nly incrpraes he inpu inefficiency and he cnribuin f inefficiency ha is made by marke prices (marke inefficiency is n cnsidered. T slve hese prblems he alernae CE mdel was presened in Tne (2002 by cnsidering he prducin echnlgy in a cs/inpu space. Fllwing he presened CE measure, Fukuyama and Weber (2004 and Färe and Grsskpf (2006 develped a direcinal inpu-cs disance funcin and herefre a direcinal measure f value-based echnical inefficiency. This measure was exended in Sah e al. (2014. They develped a direcinal value-based measure f echnical efficiency and als a direcinal cs-based measure f efficiency which saisfied he prperies: ranslain invarian, uni invarian and srng mnnciy. In his paper we esimae a direcinal measure f cs Malmquis index (DCM
Prduciviy changes f unis: A direcinal measure f cs Malmquis index 57 based n he cs and echnical efficiency measure presened in Sah e al. (2014, in rder examine he prduciviy changes f unis in siuain where DMUs ac in a nn-cmpeiive marke ha inpus are heergeneus and DMUs can cnrl sme exen he marke prices. In fac, when he inpu price vecrs f DMUs are differen because f he differen cmpeiive envirnmens, cmparing he prduciviy changes f DMUs can n be righ. Using he prpsed index he envirnmenal facrs can be incrpraed in he cmparisn beween DMUs. In addiin, DMUs can cnrl heir prduciviy changes by cnsidering he suiable direcin vecr. The res f paper prceeds as fllws. Secin 2 expresses he previus sudies n efficiency and prduciviy change when inpu prices are knwn. Secin 3 develps a cs Malmquis index fr evaluaing DMUs in a nn-cmpeiive marke. In secin 4 we design a simple numerical example shw he applicain f he prpsed apprach and Secin 5 cncludes. 2. Cs Malmquis prduciviy index Suppse ha here are n DMUs, bserved in ime perid ( 1,, T, each DMU j ( j =1,, n cnsumes a nn- negaive inpu vecr x j ( x 1 j,, x mj wih he price vecr c j ( c1 j,, c mj prduce a nn-negaive upu vecr y,, y. Farrell (1957 defined j 1 j sj he cs efficiency f a DMU as he rai f minimum cs f prducin he bserved cs. This definiin f cs efficiency requires ha he inpu prices be fixed and he exac infrmain f hem is a hand. By using he cncep f Farrell's cs efficiency, Maniadakis and Thanassulis (2004 presened he cs Malmquis prduciviy index, which evaluaes he prduciviy change f DMUs beween ime perids and 1in he case where he inpu price vecr is knwn. They assumed ha all DMUs face he same inpu price vecrs. Cnsider c c1 c m (,, as he inpu price vecr f perid ( c c, j, and define he j prducin echnlgy f perid as T = {(x, y R : x R canprduce y R } (1 The Farrell cs efficiency measure fr ( x, y, bserved in perid, under he inpu price vecr c is defined as C, c CE, x, c, (2 c x
G. Thidi, e al /JNRM Vl.1, N.2, Summer 2015 58 Where C, c is he minimum prducin cs and cs f prducing c x is he bserved y. C, c can be bained by slving he fllwing linear prgramming mdel: m i i i 1 C, c min c x, (3 J s.. x x, i 1,, m, j 1 J j 1 j ij i y y, r 1,, s, j rj r 0, j 1,, J, j x 0, i 1,, m. i Based n he cs efficiency defined in (2, and by cnsidering ime perid as he reference perid, he cs Malmquis prduciviy index ( CM is (Maniadakis and Thanassulis, 2004: c x C, c CM. (4 CM 1 1 c x C, c 1 1 ( x, y cmpares he cs efficiency f, under evaluain DMU bserved in ime perid 1, ha f ( x, y by measuring heir disances frm he cs bundary f perid as a benchmark, where he cs bundary is defined as Is C, c x : c x C, c. (5 Similarly, 1 CM index can be defined based n he cs bundary f perid 1 as a benchmark, w x C, w (6 w x C, w 1 1 1 1 1 1 CM. 1 1 1 1 CM 1 1 ( x, y cmpares he cs efficiency f ha f ( x, y by measuring heir disances frm he cs bundary f perid 1. Because w indexes CM and CM 1 may prvide differen measures f prduciviy change (he disances are cmpued based n differen benchmarks, Maniadakis and Thanassulis (2004 defined he CM index as he gemeric mean f hese w indexes as fllws: If he CM, (, = w x /C (y, w / w x /C (y, w w x /C (y, w (7 CM index has a value less han 1, prduciviy prgress, a value greaer han 1 means ha prduciviy regress and a value f 1 means ha prduciviy remains unchanged. Using he CM index examine he prduciviy change f DMUs, hey culd incrprae allcaive efficiency in he measuremen f prduciviy change. They
Prduciviy changes f unis: A direcinal measure f cs Malmquis index 59 decmpsed he presened index in cs efficiency and cs echnical change. In addiin, hey decmpsed each f hese cmpnens in w cmpnens. Cs efficiency was decmpsed in echnical and allcaive efficiency change, and cs echnical change in a par capuring shifs f inpu quaniies and shifs f inpu prices. Farrell's CE mdel used fr cmpuing he CM index requires ha he inpu price vecr f DMUs are fixed and exgenusly given. In fac, his mdel is applied fr evaluaing he cs efficiency f DMUs in a cmpeiive marke. In ms f he real applicain, we deal wih he cases where he marke is n fully cmpeiive and inpu prices may vary beween DMUs. In such siuain Farrell CE measure can n reflec he marke inefficiency. In he nex secin we sugges a direcinal cs Malmquis index fr use in siuains where DMUs peraes in a nncmpeiive marke characerized by heergeneus inpus. 3. A direcinal measure f cs Malmquis index Cs efficiency measure and cs Malmquis index discussed in he frmer secin can be applied fr he case where DMUs are hmgenus and inpu prices are exgenusly fixed (ha is DMUs are price aker. T esimae he cs efficiency f DMUs in nn-cmpeiive marke, Sah e al. (2014 develped a direcinal cs based measure f efficiency (DCE and als direcinal value based measure f echnical efficiency (TE which saisfy hree impran prperies, ranslain invariance, srng mnniciy and uni invariance if he unis f measuremen fr each cmpnen f he seleced direcin vecr g ( g, g, s g ( g, g,, g, g 0 R, be he x 1 2 m same as ha f ih inpu-cs, y x i x y. They assumed ha inpus are heergeneus and heir prices vary acrss DMUs. In rder incrprae hese assumpins in he mdel, hey defined he prducin echnlgy as T = {(x, y R : x R can prduce (8 y R } where x c x. Their presened mdel evaluae he DCE measure f DMU bserved in perid based n he prducin echnlgy f perid is as fllws: DCE (y, x min 1 g G β,
G. Thidi, e al /JNRM Vl.1, N.2, Summer 2015 60 λ x x β, g, i = 1,, m λ y y, r = 1,, s (9 λ = 1, λ 0, j = 1,, n Where G m g and guaranee i 1 ix ha DCE, x 1, he direcin vecr g mus saisfy he fllwing cndiin: x i min x ij j 1, n max 1. g ix i 1, m (7 Similarly, he DCE measure f DMU bserved in ime perid 1 wih respec he echnlgy f perid 1is, DCE (y, x min 1 g G β λ, x x β, g, i = 1,, m λ y y, r = 1,, s (11 n λ j = 1, λ j 0, j = 1,, n j1 T measure mdify DCE and DCE 1,, we Errr! Reference surce n fund. in he fllwing mdels, respecively: DCE (y, x min 1 g G β λ x, x β, g, i = 1,, m λ y y, r = 1,, s (12 λ = 1, λ 0, j = 1,, n DCE (y, x min 1 g G β λ, x x β, g, i = 1,, m λ y y, r = 1,, s (13 λ = 1, λ 0, j = 1,, n Nw, we define he direcinal cs Malmquis (DCM prduciviy index f, 1and heir gemeric mean respecively as fllws: 1 1 DCE, x DCE, x 1 1 1 1 DCE, x 1 DCE, x DCM, (8 DCM, (9 DCM, = If he,, (, / (16 (, DCM index has a value less han 1, prduciviy regress, a value greaer han 1 means ha prduciviy prgress and a value f 1 means ha prduciviy remains unchanged. The prpsed prduciviy index Errr! Reference surce n fund. is incrpraed wih he decisin maker's preferences. Therefre, imprve he prduciviy f DMUs decisin makers can selec a specific direcin vecr. In fac,
Prduciviy changes f unis: A direcinal measure f cs Malmquis index 61 he prduciviy change f DMUs can be measured based n heir specific direcin vecr. In addiin, he measure f prduciviy change bained frm Errr! Reference surce n fund. is ranslain invarian and can be used in siuains dealing wih negaive daa. 3.1. Decmpsiin f he prpsed index Nw, we decmpse he DCM index illusrae hw he index includes direcinal echnical and allcaive efficiency changes, shif f he prducin bundary beween perids and 1, and als he effec f inpu price change n he prduciviy change f DMU beween ime perids and 1. The decmpsiin is similar he decmpsiin f CM index presened in Maniadakis and Thanassulis (2004. In he firs sage, DCM index can be decmpsed in verall (cs efficiency change (OEC and cs echnical change (CTC as fllws: DCM, = DCE (y, x DCE (y, x DCE (y, x DCE (y, x / DCE (y, x DCE (y, x OEC cmpnen examine wheher he bserved cs f prducing he given (17 upu vecr, m i 1 x i, caches up he minimum cs f prducing i frm perid perid 1. Using he pimal sluin f mdel Errr! Reference surce n fund., he minimum cs f prducing calculaed as, m m *, * x i x i i g ix i 1 i 1 y can be (. (10 Similarly, he minimum cs f prducing 1 y can be calculaed by he pimal sluin f mdel Errr! Reference surce n fund.. CTC cmpnen cmpares he minimum cs f prducing he given upu vecr bserved in perid wih ha f perid 1. In he secnd sage f he decmpsiin f DCM index, each cmpnen bained in sage 1 can be furher decmpsed in w cmpnens. OEC cmpnen is decmpsed as OEC = 1 β, 1 β, DCE (y, x /(1 β, DCE (y, x /(1 β, = TEC AEC (19 ha, is he pssible echnical imprvemen f he cmpnens f inpu-
G. Thidi, e al /JNRM Vl.1, N.2, Summer 2015 62 spending x (direcinal value-based measure f echnical inefficiency. The value f, can be cmpued by he pimal sluin f mdel Errr! Reference surce n fund. as fllws:, g, 1 min 1 min.(11 m, ix i i i 1,, m i 1,, m i 1 G, can be calculaed als by slving he fllwing mdel direcly: β, = max, β, λ x x λ y (21 β, g, i = 1,, m y, r = 1,, s λ = 1, λ 0, j = 1,, n Similarly, 1, 1 can be calculaed by he pimal sluin f mdel Errr! Reference surce n fund., r direcly by slving mdel Errr! Reference surce n fund. afer replacing ime wih ime 1. The cmpnen CTC bained in he firs sage f decmpsiin can be decmpsed in TC and PE cmpnens as fllws: CTC / = 1 β, 1 β, 1 β, 1 β, DCE (y, x /(1 β, DCE (y, x /(1 β, DCE (y, x /(1 β, DCE (y, x /(1 β, = TC PE / (12, 1 The value f can be calculaed by he pimal sluin f mdel Errr! Reference surce n fund. as m, 1, 1, 1 g ix 1 min i 1 min i i 1,, m G i 1,, m i 1 r slving he fllwing mdel direcly: β, = max, β, λ x x β, g, i = 1,, m λ y y, r = 1,, s (23 λ = 1, λ 0, j = 1,, n The value f similar perids and 1. 1, can be esimaed, 1 afer changing rund he TC cmpnen esimaes he shif f he prducin bundary frm perid perid 1. PE cmpnen esimaes he effec f inpu price changes n changes f he minimum cs f prducing he given upu vecr. Fr each f he cmpnens f he DCM index, a value less han 1 indicaes regress, a value greaer 1
Prduciviy changes f unis: A direcinal measure f cs Malmquis index 63 indicaes prgress and a value f 1 express he perfrmance remains unchanged. 4. Numerical example In rder illusrae he abiliy f he prpsed apprach we have analyzed 5 DMUs wih w inpus and w upus. Table 1 shws he inpu/upu daa and he inpu price vecrs fr 5 DMUs bserved in w ime perids 0 and 1. We apply he index Errr! Reference surce n fund. and als he index Errr! Reference surce n fund. evaluae he prduciviy changes f DMUs beween perids 0 and 1. We cmpue he index Errr! Reference surce n fund. by cnsidering w direcin vecrs as g = g, g, g = x, g = 0, i = 1,, m, r = 1,, s (24 g = g, g, g = max,, x, g = 0, i = 1,, m, r = 1,, s (25 Ne ha in perid 2 DMU1 increases is inpu quaniies and simulaneusly decreases is upu quaniies while he inpu prices vecr remains unchanged frm ime 0 ime1. Therefre we expec ha prduciviy f DMU1 regress frm ime 0 ime1. I can als be derived frm Tables 6 and 7. These Tables respecively shw he resuls bained frm he indexes Errr! Reference surce n fund. and Errr! Reference surce n fund.. Tw indexes repr a regress in prduciviy fr DMU1. Frm he resuls f Table 3 i can be seen ha he amun f regress in prduciviy bained frm selecing he direcin vecr Errr! Reference surce n fund. is mre han ha f he direcin Errr! Reference surce n fund.. I means ha, DCM he decisin maker's preferences. is incrpraed wih Nw cnsider DMU5 as DMU under evaluain. This DMU imprves is upus wihu any changes in is inpus quaniies and prices. Thus we expec ha is prduciviy imprves frm ime 0 ime 1. Table 1. Numerical example daa =0 =1 MU I1 I2 C1 C2 O1 O2 I1 I2 C1 C2 O1 O2 DMU1 5 3 3 1 2 3 15 6 3 1 1 1.5 DMU2 9 5 3 1 5 4 4.5 2.5 1.5 0.5 15 12 DMU3 13 6 4 2 3 6 13 6 4 2 3 6 DMU4 15 14 2 3 7 9 15 14 10 15 7 9 DMU5 7 11 5 1 5 9 7 11 5 1 20 36
G. Thidi, e al /JNRM Vl.1, N.2, Summer 2015 64 I can be seen ha frm Tables 2 and 3, he indexes CM and DCM differen resuls fr DMU5. The prvide he CM index shws a regress in he prduciviy while he DCM index reprs an imprvemen fr he prduciviy change beween w ime perids. In addiin, he value f direcin DCM bained based n vecr Errr! Reference surce n fund. indicaes higher prduciviy grwh han he vecr DCM index based n direcin Errr! Reference surce n fund.. Therefre, i seems ha he resuls bained frm he reasnable han ha f he DCM index are mre CM index. The resuls fr he her DMUs and als he value f he DCM index cmpnens can be inerpreed similarly. In a nn-cmpeiive marke characerized by heergeneus inpus where DMUs have he abiliy influence smewha he marke prices, he envirnmenal facrs may affec n decisins f DMUS in specifying heir inpu price vecrs. In such siuains he bained resuls f cmparing he prduciviy changes f DMUs using he cs Malmquis index presened in Maniadakis and Thanassulis (2004 can n be righ. In The curren sudy we assumed ha he inpu prices are varying beween DMUs and esimaed a direcinal measure f cs Malmquis index by cnsidering he affecs f hese envirnmenal facrs n he prduciviy changes ver ime. Als, using he new cs malmquis index, decisin maker's preference can be incrpraed in he prduciviy changes f unis by selecing he suiable direcin vecr. 5. Cnclusin Table 2. The resuls f CM index DMU DMU1 DMU2 DMU3 DMU4 DMU5 CM 0,1 2.83 1.87 1.00 1.00 2.36 Table 3. The resuls f DCM index Direcin vecr 1 Direcin vecr 2 Decmpsiin f DCM index Decmpsiin f DCM index DMU DCM 0,1 TEC AEC TC PE DCM 0,1 TEC AEC TC PE DMU1 0.35 0.21 0.75 2.31 0.97 0.95 0.98 0.9 1.03 1.05 DMU 2 6 1 1 11.59 0.52 1.26 1 1 1.15 1.1 DMU 3 1 0.22 1.12 4.49 0.89 1.7 1.08 1.19 1.1 1.2 DMU 4 0.2 0.05 0.49 4.44 2.03 0.12 0.05 0.49 2.84 1.87 DMU 5 3 1 1 4.45 0.67 1.34 1 1 1.22 1.1
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