数据包络分析工具箱 EDA 算法分析

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
function [ out ] = dea( X, Y, varargin)
%DEA Data envelopment analysis radial and directional model
%   Computes data envelopment analysis radial and directional model
%
%   out = DEA(X, Y, Name, Value) computes data envelopment analysis model
%   with inputs X and outputs Y. Model properties are specified using 
%   one or more Name ,Value pair arguments.
%
%   Additional properties:
%   - 'orient': orientation. Input oriented 'io', output oriented 'oo', 
%   directional distane function 'ddf'.
%   - 'rts': returns to scale. Constant returns to scale 'crs', variable
%   returns to scale 'vrs'.
%   - 'Gx': input directions for 'ddf' orientation. Default is Xeval.
%   - 'Gy': output directions for 'ddf' orientation. Default is Yeval.
%   - 'names': DMU names.
%   - 'secondstep': 1 to compute input and output slacks. Default is 1.
%
%   Advanced parameters:
%   - 'Xeval: inputs to evaluate if different from X.
%   - 'Yeval': outputs to evaluate if different from Y.
%
%   Example
%     
%      io = dea(X, Y, 'orient', 'io');
%      oo_vrs = dea(X, Y, 'orient', 'oo', 'rts', 'vrs');
%      ddf = dea(X, Y, 'ddf', 'Gx', X, 'Gy', Y);
%
%   See also DEAOUT, DEASCALE, DEAMALM, DEAADDIT, DEASUPER
%
%   Copyright 2016 Inmaculada C. 脕lvarez, Javier Barbero, Jos茅 L. Zof铆o
%   http://www.deatoolbox.com
%
%   Version: 1.0
%   LAST UPDATE: 1, August, 2017
%

    % Check size
    if size(X,1) ~= size(Y,1)
        error('Number of rows in X must be equal to number of rows in Y')
    end    
    
    % Get number of DMUs (n), inputs (m) and outputs (s)
    [n, m] = size(X);
    s = size(Y,2);
    
    % Get DEA options
    options = getDEAoptions(n, varargin{:});
    
    % Orientation
    switch(options.orient)
        case {'io','input'}
            orient = 'io';
        case {'oo','output'}
            orient = 'oo';
        case {'ddf'}
            orient = 'ddf';
        case {'none'}
            error('Radial Model is Oriented');
    end       
    
    % RETURNS TO SCALE
    rts = options.rts;
    switch(rts)
        case 'crs'
            AeqRTS1 = [];
            beqRTS1 = [];
            
            AeqRTS2 = [];
            beqRTS2 = [];
        case 'vrs'
            AeqRTS1 = [ones(1,n), 0];
            beqRTS1 = 1;
            
            AeqRTS2 = [ones(1,n), zeros(1,m), zeros(1,s)];
            beqRTS2 = 1;
    end
    
    % If evaluate DMU at different X or Y
    if ~isempty(options.Xeval)
        Xeval = options.Xeval;
    else
        Xeval = X;
    end
    
    if ~isempty(options.Yeval)
        Yeval = options.Yeval;
    else
        Yeval = Y;
    end
    
    if size(Xeval,1) ~= size(Yeval,1)
        % Check size: rows
        error('Number of rows in Xeval and Yeval must be equal')
    end
    
    if size(Xeval,2) ~= size(X,2)
        % Check columns Xref
        error('Number of columns in Xeval and X must be equal')
    end
    
    if size(Yeval,2) ~= size(Y,2)
        % Check columns Yref
        error('Number of columns in Yeval and Y must be equal')
    end
    
    neval = size(Xeval,1);

    % OPTIMIZATION OPTIONS:
    optimopts = options.optimopts;
    
    % Create variables to store results
    lambda = nan(neval, n);
    slackX = nan(neval, m);
    slackY = nan(neval, s);
    eff = nan(neval, 1);
    Eflag = nan(neval, 2);
    Xeff = nan(neval, m);
    Yeff = nan(neval, s);
    dualineqlin = nan(neval, m + s);
    dualeqlin = nan(neval, 1);
    
    % OPTIMIZE: SOLVE LINEAR PROGRAMMING MODEL
    switch(orient)
        case 'io'
            
            % For each DMU
            for j=1:neval
                
                % FIRST STEP:
                % Objective function
                f = [zeros(1,n), 1];
                
                % Constraints
                A = [ X', -Xeval(j,:)';
                     -Y', zeros(s,1)];
                b = [zeros(m,1);
                    -Yeval(j,:)'];
                Aeq = AeqRTS1;
                beq = beqRTS1;
                lb = zeros(1, n + 1);
                                
                % Optimize
                [z, ~, exitflag, ~, dual] = linprog(f, A, b, Aeq, beq, lb, [], [], optimopts);
                if exitflag ~= 1
                    if options.warning
                        warning('DMU %i. First Step. Optimization exit flag: %i', j, exitflag)
                    end
                end
                if isempty(z)
                    if options.warning
                        warning('DMU %i. First Step. Optimization doesn''t return a result. Efficiency set to NaN.', j)
                    end
                    z = nan(n + 1, 1);  
                    dual.ineqlin = nan(1, m + s);
                    dual.eqlin = nan(1, 1);
                end
                
                % Get efficiency
                theta = z(end);
                Eflag(j, 1) = exitflag;
                eff(j) = theta;
                
                % Get dual results
                dualineqlin(j, :) = dual.ineqlin;
                if ~isempty(beqRTS1)
                    dualeqlin(j, 1) = dual.eqlin;
                else
                    dualeqlin(j, 1) = NaN;
                end                
                
                % SECOND STEP
                
                if(options.secondstep) && ~isnan(theta)
                
                    % Objective function
                    f = [zeros(1, n), -ones(1, m + s)];

                    % Constraints
                    Aeq = [ X', eye(m,m)  , zeros(m,s);
                            Y', zeros(s,m), -eye(s,s);
                           AeqRTS2];
                    beq = [theta .* Xeval(j,:)';
                            Yeval(j,:)';
                            beqRTS2];
                    lb = zeros(n + s + m, 1);

                    % Optimize
                    [z, ~, exitflag] = linprog(f, [], [], Aeq, beq, lb, [], [], optimopts);
                    if exitflag ~= 1
                        if options.warning
                            warning('DMU %i. Second Step. Optimization exit flag: %i', j, exitflag)
                        end
                    end
                    if isempty(z)
                        if options.warning
                            warning('DMU %i. Second Step. Optimization doesn''t return a result. Results set to NaN.', j)
                        end
                        z = nan(n + m + s, 1);                   
                    end

                    % Get results
                    lambda(j,:) = z(1:n);
                    slackX(j,:) = z(n + 1 : n + m);
                    slackY(j,:) = z(n + m + 1 : n + m + s);                
                    Eflag(j, 2) = exitflag;

                    % Compute efficient inputs and outputs
                    Xeff(j,:) = repmat(eff(j), 1, m) .* Xeval(j,:) - slackX(j,:);
                    Yeff(j,:) = Yeval(j,:) + slackY(j,:);

                end
                
            end
            
            % Compute efficient inputs and outputs
            % Xeff = repmat(eff, 1, m) .* Xeval - slackX;
            % Yeff = Yeval + slackY;

            
        case 'oo'
                        
            % For each DMU
            for j=1:neval
                
                % FIRST STEP:
                % Objective function (maximize)
                f = -[zeros(1,n), 1];
                
                % Constraints
                A = [ X', zeros(m,1);
                     -Y', Yeval(j,:)'];
                b = [Xeval(j,:)';
                     zeros(s,1)];
                Aeq = AeqRTS1;
                beq = beqRTS1;
                lb = zeros(1, n + 1);

                % Optimize
                [z, ~, exitflag, ~, dual] = linprog(f, A, b, Aeq, beq, lb, [], [], optimopts);
                if exitflag ~= 1
                    if options.warning
                        warning('DMU %i. First Step. Optimization exit flag: %i', j, exitflag)
                    end
                end
                if isempty(z)
                    if options.warning
                        warning('DMU %i. First Step. Optimization doesn''t return a result. Efficiency set to NaN.', j)
                    end
                    z = nan(n + 1, 1);  
                    dual.ineqlin = nan(1, m + s);
                    dual.eqlin = nan(1, 1);
                end
                
                % Get efficiency
                phi = z(end);
                eff(j) = phi;
                Eflag(j, 1) = exitflag;
                
                % Get dual results
                dualineqlin(j, :) = dual.ineqlin;
                if ~isempty(beqRTS1)
                    dualeqlin(j, 1) = dual.eqlin;
                else
                    dualeqlin(j, 1) = NaN;
                end    
                
                % SECOND STEP
                
                if(options.secondstep)  && ~isnan(phi)
                                 
                    % Objective function
                    f = -[zeros(1, n), ones(1, m + s)];

                    % Constraints
                    Aeq = [ X', eye(m,m)  ,  zeros(m,s),  ;
                            Y', zeros(s,m), -eye(s,s)  ;                       
                           AeqRTS2];
                    beq = [Xeval(j,:)'
                           phi .* Yeval(j,:)';
                           beqRTS2];
                    lb = zeros(n + s + m, 1);

                    % Optimize
                    [z, ~, exitflag] = linprog(f, [], [], Aeq, beq, lb, [], [], optimopts);
                    if exitflag ~= 1
                        if options.warning
                            warning('DMU %i. Second Step. Optimization exit flag: %i', j, exitflag)
                        end
                    end
                    if isempty(z)
                        if options.warning
                            warning('DMU %i. Second Step. Optimization doesn''t return a result. Results set to NaN.', j)
                        end
                        z = nan(n + m + s, 1);                   
                    end

                    % Get results
                    lambda(j,:) = z(1:n);
                    slackX(j,:) = z(n + 1 : n + m);
                    slackY(j,:) = z(n + m + 1 : n + m + s);                
                    Eflag(j, 2) = exitflag;

                    % Compute efficient inputs and outputs
                    Xeff(j,:) = Xeval(j,:) - slackX(j,:);
                    Yeff(j,:) = repmat(eff(j), 1, s) .* Yeval(j,:) + slackY(j,:);

                end
                    
            end
            
            % Compute efficient inputs and outputs
            % Xeff = Xeval - slackX;
            % Yeff = repmat(eff, 1, s) .* Yeval + slackY;

            
        case 'ddf'
                        
            % Get directions
            Gx = options.Gx;
            Gy = options.Gy;
                        
            if length(Gx) == 1
                Gx = repmat(Gx, size(X,1), size(X,2));
            elseif size(Gx, 1) == 1
                Gx = repmat(Gx, size(X,1), 1);
            end
            
            if length(Gy) == 1
                Gy = repmat(Gy, size(Y,1), size(Y,2));
            elseif size(Gy, 1) == 1
                Gy = repmat(Gy, size(Y,1), 1);
            end
            
            if isempty(Gx)
                Gx = Xeval;
            end
            
            if isempty(Gy)
                Gy = Yeval;
            end
                        
            % For each DMU
            for j=1:neval
                
                % FIRST STEP:
                % Objective function (maximize)
                f = -[zeros(1,n), 1];
                
                % Constraints
                A = [ X', Gx(j,:)';
                     -Y', Gy(j,:)'];
                b = [ Xeval(j,:)';
                     -Yeval(j,:)'];
                Aeq = AeqRTS1;
                beq = beqRTS1;
                %lb = zeros(1, n + 1);
                lb = [zeros(1, n), -inf];
                
                % Optimize
                [z, ~, exitflag, ~, dual] = linprog(f, A, b, Aeq, beq, lb, [], [], optimopts);
                if exitflag ~= 1
                    if options.warning
                        warning('DMU %i. First Step. Optimization exit flag: %i', j, exitflag)
                    end
                end
                if isempty(z)
                    if options.warning
                        warning('DMU %i. First Step. Optimization doesn''t return a result. Efficiency set to NaN.', j)
                    end
                    z = nan(n + 1, 1);  
                    dual.ineqlin = nan(1, m + s);
                    dual.eqlin = nan(1, 1);
                end
                
                % Get efficiency
                beta = z(end);
                eff(j) = beta;
                Eflag(j, 1) = exitflag;
                
                % Get dual results
                dualineqlin(j, :) = dual.ineqlin;
                if ~isempty(beqRTS1)
                    dualeqlin(j, 1) = dual.eqlin;
                else
                    dualeqlin(j, 1) = NaN;
                end  
                
                % SECOND STEP
                
                if(options.secondstep) && ~isnan(beta)
                                 
                    % Objective function
                    f = -[zeros(1, n), ones(1, m + s)];

                    % Constraints
                    Aeq = [ X', eye(m,m)  ,  zeros(m,s),  ;
                            Y', zeros(s,m), -eye(s,s)  ;                       
                           AeqRTS2];
                    beq = [-beta .* Gx(j,:)' + Xeval(j,:)'
                            beta .* Gy(j,:)' + Yeval(j,:)';
                           beqRTS2];
                    lb = zeros(n + s + m, 1);

                    % Optimize
                    [z, ~, exitflag] = linprog(f, [], [], Aeq, beq, lb, [], [], optimopts);
                    if exitflag ~= 1
                        if options.warning
                            warning('DMU %i. Second Step. Optimization exit flag: %i', j, exitflag)
                        end
                    end
                    if isempty(z)
                        if options.warning
                            warning('DMU %i. Second Step. Optimization doesn''t return a result. Results set to NaN.', j)
                        end
                        z = nan(n + m + s, 1);                   
                    end

                    % Get results
                    lambda(j,:) = z(1:n);
                    slackX(j,:) = z(n + 1 : n + m);
                    slackY(j,:) = z(n + m + 1 : n + m + s);                
                    Eflag(j, 2) = exitflag;

                    % Compute efficient inputs and outputs
                    Xeff(j,:) = Xeval(j,:) - repmat(eff(j), 1, m) .* Gx(j,:) - slackX(j,:);
                    Yeff(j,:) = Yeval(j,:) + repmat(eff(j), 1, s) .* Gy(j,:) + slackY(j,:);
                end
                
            end

            
    end   
  
    % Slacks structure
    slack.X = slackX;
    slack.Y = slackY;  
    
    % Dual structure
    dual.X = dualineqlin(:, 1:m);
    dual.Y = dualineqlin(:, m+1: m+s);
    if ~isempty(beqRTS2)
        dual.rts = dualeqlin(:);
    else
        dual.rts = nan(neval,1);
    end
    
    % SAVE results and input data
    out = deaout('n', n, 'neval', neval', 's', s, 'm', m,...
        'X', X, 'Y', Y, 'names', options.names,...
        'model', 'radial', 'orient', orient, 'rts', rts,...
        'lambda', lambda, 'slack', slack,...
        'eff', eff, 'Xeff', Xeff, 'Yeff', Yeff,...
        'dual', dual,...
        'exitflag', Eflag,...
        'dispstr', 'names/X/Y/eff/slack.X/slack.Y');


end

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
function [ T ] = ( out, dispstr )
%DEA2TABLE Convert 'deaodea2tableut' results into a table object
%   Convert data envelopment analysis results in a 'deaout' structure into
%   a MATLAB table object.
%
%   T = DEA2TABLE( out ) Converts 'deaout' structure into a MATLAB table
%   object.
%   T = DEA2TABLE( out, dispstr ) Converts 'deaout' structure into a MATLAB
%   table object using the specified 'dispstr' structure.
%
%   Example
%       
%       io = dea(X, Y, 'orient', 'io');
%       T = dea2table(io);
%
%       T2 = dea2table(io, 'names/lambda/eff');
%
%   See also DEAOUT, DEADISP
%
%   Copyright 2016 Inmaculada C. 羖varez, Javier Barbero, Jos� L. Zof韔
%   http://www.deatoolbox.com
%
%   Version: 1.0
%   LAST UPDATE: 9, May, 2017
%

    if nargin < 2
        dispstr = out.dispstr;
    end    
    
    % Convert to table
    dispstr = strsplit(dispstr, '/');
    
    % Create empty table
    T = table();
    
    % Build Header and Body
    for i=1:length(dispstr)  
            % Get param name
            paramstr = char(dispstr(i));
            
            % Get data
            dat = eval(sprintf('out.%s', paramstr));
            
            % Append to Table
            T = [T table(dat)];
            
            % Get variable name
            [name, ~] = getDEAformat(paramstr, out.orient);
            
            % If no name in output structure
            if isempty(name)
                disptext_field = sprintf('disptext_%s', strrep(paramstr,'.','_'));
                if isfield(out, disptext_field)
                    % If custom name exists in the output structure use it
                    name = eval(sprintf('out.%s',disptext_field));
                else
                    % If not, display paramstr name without eff.
                    name = strrep(paramstr, 'eff.', '');
                end
            end
            
            % Store variable name
            T.Properties.VariableNames(size(T,2)) = cellstr(name);
            
    end
    

end