Efficiency Analysis: How to Use Data Envelopment Analysis to Compare Efficiencies of Garages

The first several PROC OPTMODEL statements declare index sets and parameters and then read the input data:

proc optmodel;
   set <str> INPUTS;
   read data inputs into INPUTS=[input];

   set <str> OUTPUTS;
   read data outputs into OUTPUTS=[output];

   set <num> GARAGES;
   str garage_name {GARAGES};
   num input  {INPUTS, GARAGES};
   num output {OUTPUTS, GARAGES};
   read data garage_data into GARAGES=[_N_] garage_name
      {i in INPUTS}  <input[i,_N_]=col(i)>
      {i in OUTPUTS} <output[i,_N_]=col(i)>;

   num k;
   num efficiency_number {GARAGES};
   num weight_sol {GARAGES, GARAGES};

The following statements correspond directly to the mathematical programming formulation that is described earlier:

   var Weight {GARAGES} >= 0;
   var Inefficiency >= 0;

   max Objective = Inefficiency;

   con Input_con {i in INPUTS}:
      sum {j in GARAGES} input[i,j] * Weight[j] <= input[i,k];

   con Output_con {i in OUTPUTS}:
      sum {j in GARAGES} output[i,j] * Weight[j] >= output[i,k] * Inefficiency;

The following statements loop over all garages, call the linear programming solver once per garage, and store the results in the parameters efficiency_number and weight_sol:

   do k = GARAGES;
      solve;
      efficiency_number[k] = 1 / Inefficiency.sol;
      for {j in GARAGES}
         weight_sol[k,j] = (if Weight[j].sol > 1e-6 then Weight[j].sol else .);
   end;

Note that the DO loop contains no declaration statements. As the value of k changes, the SOLVE statement automatically updates the constraints to use the values of input[i,k] and output[i,k]. The approach shown here is much more efficient than the following alternatives:

In SAS/OR 13.1, you can instead solve the same set of problems in parallel by replacing the DO loop with the following COFOR loop:

   cofor {kk in GARAGES} do;
      k = kk;
      solve;
      efficiency_number[k] = 1 / Inefficiency.sol;
      for {j in GARAGES}
         weight_sol[k,j] = (if Weight[j].sol > 1e-6 then Weight[j].sol else .);
   end;

After the DO or COFOR loop terminates, the following statements partition the garages into two sets by using a threshold on the resulting efficiency numbers:

   set EFFICIENT_GARAGES = {j in GARAGES: efficiency_number[j] >= 1};
   set INEFFICIENT_GARAGES = GARAGES diff EFFICIENT_GARAGES;

The following statements print the efficiency numbers, as shown in Figure 22.1, and write them to the efficiency_data data set:

   print garage_name efficiency_number;
   create data efficiency_data from [garage] garage_name efficiency_number;

The following CREATE DATA statements write the inefficient garages and the corresponding multiples of efficient garages to SAS data sets (in both dense and sparse form), as in Table 14.8 in Williams (1999):

   create data weight_data_dense from [inefficient_garage]=INEFFICIENT_GARAGES
      garage_name
      efficiency_number
      {efficient_garage in EFFICIENT_GARAGES} <col('w'||efficient_garage)
         =weight_sol[inefficient_garage,efficient_garage]>;
   create data weight_data_sparse from
      [inefficient_garage efficient_garage]=
   {g1 in INEFFICIENT_GARAGES, g2 in EFFICIENT_GARAGES: weight_sol[g1,g2] ne .}
      weight_sol;
quit;

The following statements sort the efficiency_data data set by efficiency and print the results, shown in Figure 22.2:

proc sort data=efficiency_data;
   by descending efficiency_number;
run;

proc print;
run;

The following statements sort the weight_data_dense data set by efficiency and print the results, shown in Figure 22.3:

proc sort data=weight_data_dense;
   by descending efficiency_number;
run;

proc print;
run;

The weight_data_sparse data set contains the same information in sparse format, as shown in Figure 22.4:

proc print data=weight_data_sparse;
run;