{"id":167,"date":"2024-03-25T11:45:41","date_gmt":"2024-03-25T11:45:41","guid":{"rendered":"https:\/\/dataenvelopment.com\/gams\/?p=167"},"modified":"2024-06-10T06:53:48","modified_gmt":"2024-06-10T06:53:48","slug":"chapter-7-fig7-01","status":"publish","type":"post","link":"https:\/\/dataenvelopment.com\/gams\/chapter-7-fig7-01\/","title":{"rendered":"Chapter 7 (Fig. 7.01) – The Mathematical formulation for MPI (Malmquist Productivity Index) and the corresponding GAMS code"},"content":{"rendered":"\n
$Title Chapter 7 (Fig. 7.1)\n$Title Mathematical formulation for MPI (Malmquist Productivity Index) and the corresponding GAMS code\n\n$onText\n\nIf using this code, please cite:\n\n---------------------------------------------------------------------------------\nEmrouznejad, A., P. Petridis, and V. Charles (2023). Data Envelopment Analysis\nwith GAMS: A Handbook on Productivity Analysis, and Performance Measurement,\nSpringer, ISBN: 978-3-031-30700-3.\n---------------------------------------------------------------------------------\n\nWebsite: https:\/\/dataenvelopment.com\/GAMS\/\n\n$offText\n\nSets    j DMUs \/DMU1*DMU6\/\n        t years \/Y2007, Y2008\/\n        g Inputs and Outputs \/IN1, IN2, OUT1, OUT2\/\n        i(g)  Inputs \/IN1, IN2\/\n        r(g) Outputs \/OUT1, OUT2\/;\n        alias(jj,j);\n\nTable Data(j,t,g) Data for inputs and outputs\n\n                 IN1        IN2       OUT1       OUT2\nDMU1.Y2007        15        2         14        3.5\nDMU2.Y2007        40        7         14        21\nDMU3.Y2007        32        12        42        10.5\nDMU4.Y2007        52        20        28        42\nDMU5.Y2007        35        12        19        30\nDMU6.Y2007        32        7         14        38\nDMU1.Y2008        10        1.5        17       2.5\nDMU2.Y2008        45        5.6        16       22\nDMU3.Y2008        35        11        40        10\nDMU4.Y2008        50        27        28        30\nDMU5.Y2008        30        14        19        25\nDMU6.Y2008        38        9        13         12 ;\n\n\nVariables theta1  \u00ef\u00bf\u00bd for D1 model\n          theta2  \u00ef\u00bf\u00bd for D2 model\n          theta3  \u00ef\u00bf\u00bd for D3 model\n          theta4  \u00ef\u00bf\u00bd for D4 model;\n\nNonnegative variables\n          l(j) dual weights (Lambda values);\n\nParameters DMU_data(g,t) slice of data\n           D_1(j) calculated efficiency for mixed period t+1 and t\n           D_2(j) calculated efficiency for mixed period t and t+1\n           D_3(j) calculated efficiency for period t\n           D_4(j) calculated efficiency for period t+1\n           max_t max period of time\n           Deff(j) Efficiency change for each DMU\n           Dtech(j) Technical efficiency change for each DMU\n           MPI(j) Malmquist Productivity Index for each DMU;\n\nmax_t = SMAX(t,ORD(t));\n\nEquations  CON1(i,t) Input constraint for D1 model\n           CON2(r,t) Output constraints for D1 model\n           CON3(i,t) Input constraint for D2 model\n           CON4(r,t) Output constraints for D2 model\n           CON5(i,t) Input constraint for D3 model\n           CON6(r,t) Output constraints for D3 model\n           CON7(i,t) Input constraint for D4 model\n           CON8(r,t) Output constraints for D4 model;\n\nCON1(i,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t+1,i))=L=theta1*DMU_data(i,t);\nCON2(r,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t+1,r))=G=DMU_data(r,t);\n\nCON3(i,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t,i))=L=theta2*DMU_data(i,t+1);\nCON4(r,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t,r))=G=DMU_data(r,t+1);\n\nCON5(i,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t,i))=L=theta3*DMU_data(i,t);\nCON6(r,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t,r))=G=DMU_data(r,t);\n\nCON7(i,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t+1,i))=L=theta4*DMU_data(i,t+1);\nCON8(r,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t+1,r))=G=DMU_data(r,t+1);\n\nmodel D1 \/CON1, CON2\/;\nmodel D2 \/CON3, CON4\/;\nmodel D3 \/CON5, CON6\/;\nmodel D4 \/CON7, CON8\/;\n\nloop(jj,\n   DMU_data(g,t) = Data(jj,t,g);\n   solve D1 using LP minimizing theta1;\n   D_1(jj) = theta1.l;\n   solve D2 using LP minimizing theta2;\n   D_2(jj) = theta2.l;\n   solve D3 using LP minimizing theta3;\n   D_3(jj) = theta3.l;\n   solve D4 using LP minimizing theta4;\n   D_4(jj) = theta4.l;\n   Deff(jj) = D_4(jj)\/D_3(jj);\n   Dtech(jj) = ((D_2(jj)\/D_4(jj))*(D_3(jj)\/D_1(jj)))**(0.5);\n   MPI(jj) = Deff(jj)*Dtech(jj);\n );\n\nDisplay D_1,D_2,D_3,D_4, Deff, Dtech, MPI;\n\nexecute_unload<\/pre>\n","protected":false},"excerpt":{"rendered":"
$Title Chapter 7 (Fig. 7.1) $Title Mathematical formulation for MPI (Malmquist Productivity Index) 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\/167"}],"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=167"}],"version-history":[{"count":4,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/posts\/167\/revisions"}],"predecessor-version":[{"id":234,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/posts\/167\/revisions\/234"}],"wp:attachment":[{"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/media?parent=167"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/categories?post=167"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/tags?post=167"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}