{"id":90,"date":"2023-11-10T21:28:30","date_gmt":"2023-11-10T21:28:30","guid":{"rendered":"https:\/\/dataenvelopment.com\/gams\/?p=90"},"modified":"2024-03-25T11:31:55","modified_gmt":"2024-03-25T11:31:55","slug":"chapter-3-fig-3-16","status":"publish","type":"post","link":"https:\/\/dataenvelopment.com\/gams\/chapter-3-fig-3-16\/","title":{"rendered":"Chapter 3 (Fig. 3.16) &#8211; The Mathematical Formulation Of The Input DEA Congestion Using Slack Variables With The Corresponding GAMS Formulation"},"content":{"rendered":"\n<pre class=\"wp-block-preformatted\">$Title Chapter 3 (Fig. 3.16)\r\n$Title The Mathematical Formulation Of The Input DEA Congestion Using Slack Variables With The Corresponding GAMS Formulation\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 \/ProdCost, TrnCost, \r\n                             HoldInv, SatDem, Rev, CO2\/\r\n\r\n        i(g)  Inputs under managerial control \/ProdCost, \r\n                                               TrnCost, \r\n                                               HoldInv\/\r\n        r(g) Desirable outputs \/SatDem, Rev\/;\r\n        alias(jj,j);\r\n        alias(kk,jj);\r\nTable Data(j,g) Data for inputs and outputs\r\n\r\n           ProdCost     TrnCost      HoldInv     SatDem      Rev\r\nDMU1        0.255        0.161        0.373        20        2.64\r\nDMU2        0.98         0.248        0.606        6         5.29\r\nDMU3        0.507        0.937        0.749        17        2.43\r\nDMU4        0.305        0.249        0.841        2         8.99\r\nDMU5        0.659        0.248        0.979        19        2.94\r\nDMU6        0.568        0.508        0.919        17        0.75\r\nDMU7        0.583        0.628        0.732        17        6.36\r\nDMU8        0.627        0.675        0.738        10        7.2\r\nDMU9        0.772        0.657        0.486        9         2.16\r\nDMU10       0.917        0.639        0.234        8         7.3;\r\n\r\n  Variables efficiency    objective function for model 1\r\n            efficiency1   objective function for model 2\r\n            Theta          Strong disposability efficiency\r\n            Theta_tilde    Weak disposability efficiency\r\n            zeta(i)        Congestion with slack measures\r\n            Lambda(j) dual weights (Lambda values)\r\nNonnegative Variables          \r\n          Lambda(j) dual weights (Lambda values)\r\n          sminus(i)      slack variable assigned to input\r\n          splus(r)       slack variable assigned to output;\r\nParameters DMU_data(g) slice of data\r\n             eff_Theta(j)   Strong disposability efficiency  \r\n                            values\r\n             res_eff_2(j)   results for sum of zeta variables\r\n             res_zeta(j,i)   Results for zeta variable  \r\n             Lamres_s(j,j)   peers for each DMU for strong \r\n                             disposability model  \r\n\r\n             slice_Theta     slice of efficiency\r\n             slice_xproj(i)  slice for input projection\r\n             slice_yproj(r)  slice for output projection\r\n             slice_sminus(i) slice for input slack\r\n             slack_in(j,i)   results for input slack variables\r\n             slack_out(j,r)  results for output slack variables\r\n             xproj(j,i)      projected values for inputs for \r\n                             each DMU\r\n             yproj(j,r)      projected values for outputs for \r\n                             each DMU;\r\n\r\n  Equations OBJ_s objective function for strong disposability\r\n            CON1_s(i) input constraint for strong disposability\r\n            CON2_s(r) output constraint for strong disposability\r\n            OBJ_w objective function for weak disposability\r\n            CON1_w(i) input constraint for weak disposability\r\n            CON2_w(r) output constraint for weak disposability\r\n            CON3_w(i) slack constraint\r\n            VRS    VRS constraint;\r\n            \r\n  OBJ_s..    efficiency=E=Theta-1E-3*(SUM(i,sminus(i))+SUM(r,splus(r)));   \r\n  CON1_s(i)..  SUM(j, Lambda(j)*Data(j,i))=L=Theta*DMU_data(i);\r\n  CON2_s(r)..  SUM(j, Lambda(j)*Data(j,r))=G=DMU_data(r);\r\n  OBJ_w..       efficiency1=E=SUM(i,zeta(i));\r\n  CON1_w(i)..  SUM(j, Lambda(j)*Data(j,i))-\r\n             zeta(i)=E=slice_xproj(i);  \r\n  CON2_w(r)..  SUM(j, Lambda(j)*Data(j,r))=E=slice_yproj(r);\r\n  VRS.. SUM(j, Lambda(j))=E=1;\r\n  model Input_DEA_strong_disposability Input oriented DEA model for strong disposability \/OBJ_s, CON1_s, CON2_s, VRS\/;\r\n  model Input_DEA_weak_disposability_slack  Input oriented DEA model for weak disposability with slack measures \/OBJ_w, CON1_w, \r\n                                         CON2_w, CON3_w, VRS\/;\r\n loop(jj,\r\n   DMU_data(g) = Data(jj,g);\r\n   \r\n       solve Input_DEA_strong_disposability using LP \r\n             minimizing Theta;\r\n       eff_Theta(jj)=Theta.l;\r\n       slack_in(jj,i)=sminus.l(i);\r\n       slack_out(jj,r)=splus.l(r);\r\n       xproj(jj,i)=eff_theta(jj)*Data(jj,i)-\r\n                   slack_in(jj,i);\r\n       yproj(jj,r)=Data(jj,r)+slack_out(jj,r);\r\n       slice_xproj(i)=xproj(jj,i);\r\n       slice_yproj(r)=yproj(jj,r);\r\n       slice_Theta=eff_Theta(jj);\r\n       slice_sminus(i)=slack_in(jj,i);\r\n       solve Input_DEA_strong_disposability using LP minimizing Theta;\r\n       eff_Theta(jj)=Theta.l;\r\n      slack_in(jj,i)=sminus.l(i); slack_out(jj,r)=splus.l(r);\r\n      \r\n    xproj(jj,i)=eff_theta(jj)*Data(jj,i)-slack_in(jj,i); yproj(jj,r)=Data(jj,r)+slack_out(jj,r);\r\n    slice_xproj(i)=xproj(jj,i); slice_yproj(r)=yproj(jj,r); slice_Theta=eff_Theta(jj); slice_sminus(i)=slack_in(jj,i);\r\n    \r\n    solve Input_DEA_weak_disposability_slack using LP maximizing efficiency1;\r\n    res_zeta(jj,i)=zeta.l(i);\r\n    res_eff_2(jj)=SUM(i,zeta.l(i));\r\n    );\r\n    Display eff_Theta, res_zeta, res_eff_2;\r\n    \r\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>$Title Chapter 3 (Fig. 3.16) $Title The Mathematical Formulation Of The Input DEA Congestion Using Slack Variables With The Corresponding GAMS Formulation $onText If using this code, please cite: &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; 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. &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212;&#8212; [&hellip;]<\/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\/90"}],"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=90"}],"version-history":[{"count":2,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/posts\/90\/revisions"}],"predecessor-version":[{"id":160,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/posts\/90\/revisions\/160"}],"wp:attachment":[{"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/media?parent=90"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/categories?post=90"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dataenvelopment.com\/gams\/wp-json\/wp\/v2\/tags?post=90"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}