{"id":173,"date":"2024-03-25T11:51:09","date_gmt":"2024-03-25T11:51:09","guid":{"rendered":"https:\/\/dataenvelopment.com\/gams\/?p=173"},"modified":"2024-06-10T06:51:33","modified_gmt":"2024-06-10T06:51:33","slug":"chapter-7-fig-7-08","status":"publish","type":"post","link":"https:\/\/dataenvelopment.com\/gams\/chapter-7-fig-7-08\/","title":{"rendered":"Chapter 7 (Fig. 7.08) – The Mathematical formulation for MLPI (Malmquist\u00e2\u20ac\u201cLuenberger Productivity Index) and the corresponding GAMS code"},"content":{"rendered":"\n
$Title Chapter 7 (Fig. 7.8)\n$Title Mathematical formulation for MLPI (Malmquist\u00e2\u20ac\u201cLuenberger 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, UND\/\n        i(g)  Inputs \/IN1, IN2\/\n        r(g) Desirable Outputs \/OUT1, OUT2\/\n        k(g) Undesirable Outputs \/UND\/;\n        alias(jj,j);\n\nTable Data(j,t,g) Data for inputs and outputs\n\n                 IN1        IN2       OUT1      OUT2  UND\nDMU1.Y2007        15        2         14        3.5    5\nDMU2.Y2007        40        7         14        21     6\nDMU3.Y2007        32        12        42        10.5   10\nDMU4.Y2007        52        20        28        42     4\nDMU5.Y2007        35        12        19        30     3\nDMU6.Y2007        32        7         14        38     11\nDMU1.Y2008        10        1.5       17        2.5    15\nDMU2.Y2008        45        5.6       16        22     10\nDMU3.Y2008        35        11        40        10      5\nDMU4.Y2008        50        27        28        30     11\nDMU5.Y2008        30        14        19        25      2\nDMU6.Y2008        38        9         13        12      3;\n\n\nVariables beta1  \u00ef\u00bf\u00bd for D1 model\n          beta2  \u00ef\u00bf\u00bd for D2 model\n          beta3  \u00ef\u00bf\u00bd for D3 model\n          beta4  \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           MLPI_D_1(j) calculated efficiency for mixed period t+1 and t\n           MLPI_D_2(j) calculated efficiency for mixed period t and t+1\n           MLPI_D_3(j) calculated efficiency for period t\n           MLPI_D_4(j) calculated efficiency for period t+1\n           max_t max period of time\n           MPLI_Deff(j) Efficiency change for each DMU\n           MPLI_Dtech(j) Technical efficiency change for each DMU\n           MLPI(j) Malmquist Luenberger 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) Desirable Output constraints for D1 model\n           CON3(k,t) Undesirable Output constraints for D1 model\n\n           CON4(i,t) Input constraint for D2 model\n           CON5(r,t) Desirable Output constraints for D2 model\n           CON6(k,t) Undesirable Output constraints for D2 model\n\n           CON7(i,t) Input constraint for D3 model\n           CON8(r,t) Desirable Output constraints for D3 model\n           CON9(k,t) Undesirable Output constraints for D3 model\n\n           CON10(i,t) Input constraint for D4 model\n           CON11(r,t) Desirable Output constraints for D4 model\n           CON12(k,t) Undesirable Output constraints for D4 model\n;\n\n\nCON1(i,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t+1,i))=L=DMU_data(i,t);\nCON2(r,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t+1,r))=G=(1+beta1)*DMU_data(r,t);\nCON3(k,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t+1,k))=E=(1-beta1)*DMU_data(k,t);\n\nCON4(i,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t,i))=L=DMU_data(i,t+1);\nCON5(r,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t,r))=G=(1+beta2)*DMU_data(r,t+1);\nCON6(k,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t,k))=E=(1-beta2)*DMU_data(k,t+1);\n\n\nCON7(i,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t,i))=L=DMU_data(i,t);\nCON8(r,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t,r))=G=(1+beta3)*DMU_data(r,t);\nCON9(k,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t,k))=E=(1-beta3)*DMU_data(k,t);\n\nCON10(i,t)$(ORD(t)<max_t)..  SUM(j, l(j)*Data(j,t+1,i))=L=DMU_data(i,t+1);\nCON11(r,t)$(ORD(t)<max_t).. SUM(j, l(j)*Data(j,t+1,r))=G=(1+beta4)*DMU_data(r,t+1);\nCON12(k,t)$(ORD(t)<max_t).. SUM(j, l(j)*Data(j,t+1,k))=E=(1-beta4)*DMU_data(k,t+1);\n\n\nmodel MLPI_D1 \/CON1, CON2, CON3\/;\nmodel MLPI_D2 \/CON4, CON5, CON6\/;\nmodel MLPI_D3 \/CON7, CON8, CON9\/;\nmodel MLPI_D4 \/CON10, CON11, CON12\/;\n\nloop(jj,\n   DMU_data(g,t) = Data(jj,t,g);\n   solve MLPI_D1 using LP maximizing beta1;\n   MLPI_D_1(jj) = 1+beta1.l;\n   solve MLPI_D2 using LP maximizing beta2;\n   MLPI_D_2(jj) = 1+beta2.l;\n   solve MLPI_D3 using LP maximizing beta3;\n   MLPI_D_3(jj) = 1+beta3.l;\n   solve MLPI_D4 using LP maximizing beta4;\n   MLPI_D_4(jj) = 1+beta4.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 MLPI_D_1, MLPI_D_2, MLPI_D_3, MLPI_D_4;\n\nexecute_unload<\/pre>\n","protected":false},"excerpt":{"rendered":"
$Title Chapter 7 (Fig. 7.8) $Title Mathematical formulation for MLPI (Malmquist\u00e2\u20ac\u201cLuenberger 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\/173"}],"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=173"}],"version-history":[{"count":3,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/posts\/173\/revisions"}],"predecessor-version":[{"id":233,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/posts\/173\/revisions\/233"}],"wp:attachment":[{"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/media?parent=173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/categories?post=173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/tags?post=173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}