{"id":143,"date":"2023-11-10T22:12:44","date_gmt":"2023-11-10T22:12:44","guid":{"rendered":"https:\/\/dataenvelopment.com\/gams\/?p=143"},"modified":"2023-11-10T22:13:46","modified_gmt":"2023-11-10T22:13:46","slug":"chapter-5-fig-5-01","status":"publish","type":"post","link":"https:\/\/dataenvelopment.com\/gams\/chapter-5-fig-5-01\/","title":{"rendered":"Chapter 5 (Fig. 5.01) – The mathematical formulation for Allocative and Cost Efficiency"},"content":{"rendered":"\n
$Title Chapter 5 (Fig. 5.1)\r\n$Title Mathematical formulation for Allocative and Cost Efficiency and the corresponding GAMS code\r\n\r\n$onText\r\n\r\nIf using this code, please cite:\r\n\r\n---------------------------------------------------------------------------------\r\nEmrouznejad, A., P. Petridis, and V. Charles (2023). Data Envelopment Analysis\r\nwith GAMS: A Handbook on Productivity Analysis, and Performance Measurement,\r\nSpringer, ISBN: 978-3-031-30700-3.\r\n---------------------------------------------------------------------------------\r\n\r\nWebsite: https:\/\/dataenvelopment.com\/GAMS\/\r\n\r\n$offText\r\n\r\nSets    j DMUs \/DMU1*DMU10\/\r\n        g Inputs and Outputs \/Prodc, Trn, Inv, SatDem, Rev\/\r\n        i(g)  Inputs \/Prodc, Trn, Inv\/\r\n        r(g) Outputs \/SatDem, Rev\/;\r\n        alias(jj,j);\r\n        alias(k,jj);\r\n\r\nTable Data(j,g) Data for inputs and outputs\r\n\r\n            Prodc     Trn      Inv        SatDem        Rev\r\nDMU1        10        100       61           20          2.64\r\nDMU2        52        125       100          6          5.29\r\nDMU3        24         54       56           17         2.43\r\nDMU4        45         91       14            2         8.99\r\nDMU5        51         10       67           19         2.94\r\nDMU6        52         26       56           17         0.75\r\nDMU7        22         35       34           17         6.36\r\nDMU8        91         56       101          10         7.2\r\nDMU9        43         72       55            9         2.16\r\nDMU10       34         39       16            8         7.3;\r\n\r\n\r\nTable Cost(j,i) Cost Data for inputs\r\n\r\n            Prodc         Trn          Inv\r\nDMU1        0.255        0.161        0.373\r\nDMU2        0.98         0.248        0.606\r\nDMU3        0.507        0.937        0.749\r\nDMU4        0.305        0.249        0.841\r\nDMU5        0.659        0.248        0.979\r\nDMU6        0.568        0.508        0.919\r\nDMU7        0.583        0.628        0.732\r\nDMU8        0.627        0.675        0.738\r\nDMU9        0.772        0.657        0.486\r\nDMU10       0.917        0.639        0.234;\r\n\r\n\r\nVariables efficiency objective function\r\n          cost_efficiency cost efficiency\r\n          Theta     efficiency (Theta values);\r\n\r\nNonnegative variables\r\n          l(j) dual weights (Lambda values)\r\n          x_t(i) auxilliary variable for c*theta;\r\n\r\nParameters DMU_data(g) slice of data\r\n           cst(i) slice of cost data\r\n           eff(j) efficiency\r\n           tech_eff(j) technical efficiency\r\n           cost_eff(j) cost efficiency\r\n           alloc_eff(j) allocative efficiency\r\n           lamres(j,j) peers for each DMU;\r\n\r\nEquations OBJ_INP objective function for input VRS model\r\n          OBJ_COST objective function for cost efficiency model\r\n          CON1_COST(i) inputs for cost efficiency model\r\n          CON1_INP(i) input duals for input VRS model\r\n          CON2_INP(r) output dual for input VRS model\r\n          CON3 VRS orientation;\r\n\r\nOBJ_INP..       efficiency=E=Theta;\r\n\r\nCON1_INP(i)..  SUM(j, l(j)*Data(j,i))=L=Theta*DMU_data(i);\r\n\r\nCON2_INP(r)..  SUM(j, l(j)*Data(j,r))=G=DMU_data(r);\r\n\r\nCON3..         SUM(j, l(j))=E=1;\r\n\r\nOBJ_COST..       cost_efficiency=E=SUM(i,cst(i)*x_t(i));\r\n\r\nCON1_COST(i)..   SUM(j, l(j)*Data(j,i))=L=x_t(i);\r\n\r\nmodel INPUT_DEA_VRS input oriented DEA CRS \/ OBJ_INP, CON1_INP, CON2_INP, CON3\/;\r\nmodel COST_DEA_VRS input oriented DEA CRS \/ OBJ_COST, CON1_COST, CON2_INP, CON3\/;\r\n\r\n\r\nloop(jj,\r\n   DMU_data(g) = Data(jj,g);\r\n   cst(i) = Cost(jj,i);\r\n   solve INPUT_DEA_VRS using LP minimizing Theta;\r\n   solve COST_DEA_VRS using LP minimizing cost_efficiency;\r\n   cost_eff(jj) = SUM(i,cst(i)*x_t.l(i))\/SUM(i,cst(i)*DMU_data(i));\r\n   eff(jj)=Theta.l;\r\n   tech_eff(jj)= eff(jj);\r\n   alloc_eff(jj) = cost_eff(jj)\/tech_eff(jj);\r\n   loop(k,\r\n      Lamres(jj,k)=l.l(k);\r\n    );\r\n);\r\n\r\ndisplay  eff, lamres, tech_eff, cost_eff, alloc_eff;\r\n\r\nexecute_unload<\/pre>\n","protected":false},"excerpt":{"rendered":"
$Title Chapter 5 (Fig. 5.1) $Title Mathematical formulation for Allocative and Cost Efficiency and the corresponding GAMS code $onText If using this code, please cite: ——————————————————————————— Emrouznejad, A., P. Petridis, and V. Charles (2023). Data Envelopment Analysis with GAMS: A Handbook on Productivity Analysis, and Performance Measurement, Springer, ISBN: 978-3-031-30700-3. ——————————————————————————— Website: https:\/\/dataenvelopment.com\/GAMS\/ $offText Sets […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[2],"tags":[],"_links":{"self":[{"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/posts\/143"}],"collection":[{"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/comments?post=143"}],"version-history":[{"count":1,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/posts\/143\/revisions"}],"predecessor-version":[{"id":144,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/posts\/143\/revisions\/144"}],"wp:attachment":[{"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/media?parent=143"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/categories?post=143"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/tags?post=143"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}