A novel Data Envelopment Analysis model for ... - IngentaConnect

292 M. Mahdiloo et al.w o free.The dual form of the Model (1) is written as Model (2), which is known as theenvelopment additive model.Maxm s∑s− i+ ∑ s+ri= 1 r=1s.t.n∑j = 1nj = 1−x λ + s = x , i = 1, ..., m,ij j i io+∑ yrjλj− sr = yr, r = 1,..., s,(2)on∑ λj= 1, j = 1, …, n,j = 1λ j ≥ 0, j = 1, ...., n,s − ≥ 0, i = 1, ..., m,is + ≥ 0, r = 1, ..., s.rModels (1) and (2) are based on the VRS assumption. Similar to Korhonen and Luptacik(2004) and Yang and Pollitt (2009), undesirable outputs are incorporated into theModel (1) like inputs. This idea leads to a type of multiplier VRS additive model thatcan deal with undesirable outputs (Model 3). Suppose that the output variables could beDUpartitioned into subsets of desirable ( yrj) and undesirable ( yrj) outputs. Thus,O = {1,2, …, s} = O D ∪O U . In addition, there is a set of n DMUs (suppliers), DMU j :j = {1,2, …, n}, which use multiple inputs x ij (i = 1,2, …, m) to produce multipleundesirable outputs y U rj(r ∈ O U ) and desirable outputs yD rj(r ∈ O D ). The model is asfollows:MinmU U D D∑vi xi +o∑ ur yr −o∑ ur yr + wo oi= 1r∈OUr∈ODs.t.mU U D D∑ i ij ∑ r rj∑ r rjoi= 1 r∈OUr∈ODvx + u y − u y + w ≥ 0, j=1, ..., n,ν i ≥ 1, i = 1, …, m, (3)Uu ≥1, r∈O ,rUDu ≥1, r∈O ,rw o free.D