Intelligent evaluation approach for smart city based on DEA model: Taking Wuhan,China as an example

3.2.1 With the C²R model of Non Archimedes infinitesimal

Is equipped with n to participate in the appraisal unit for DMU (decision making units), each DMU has m input and s output. The J-TH decision-making unit of input vector and output vector are: X j = ( x 1 j , x 2 j , … , x mj ) T > 0(1)Y j = ( y 1 j , y 2 j , … , y sj ) T > 0 ( j = 1 , 2 , … , n )(2)

Output and input weights of indicators are: u = ( u 1 , u 2 , … , u s ) T , v = ( v 1 , v 2 , … , v m ) T .(3)

Each decision-making unit corresponding evaluation index are: h j = ∑ r = 1 s u r y rj ∑ i = 1 m v i x ij(4)

Through the use of formula (4) the efficiency evaluation index model, and the DMU efficiency index h_j0 (decision making units’ efficiency index h _j < 1,  j = 1,  2, …,  n) maximization as the goal to build the initial C²R model of DEA.

max u T Y j 0 v T X jo s . t . u T Y j ≤ 1 , j = 1 , 2 , … , n , u ≥ 0 , v ≥ 0(5)

In order to facilitate solving, transforming formula (5) to Charness-Cooper, building up an equivalent linear programming model.

If t = 1 v T X j 0 , ω = tv , μ = tu transform formula (5) to max μ ^T Y_j0s . t . { ω T X J - μ T Y j ≥ 0 , j = 1 , 2 , … , n ω T X jn = 1 ω ≥ 0 , μ ≥ 0(6)

Formulas (5) and (6) are equivalent, the optimal values are equal. And define one to define two presents DEA efficient conditions of judging decision making units DMU _jo :

Define one: if the optimal value of linear programming (6) is μ^*TY _jo = 1, so DMU _jo is called a weak DEA effective.

Define two: if the optimal value of linear programming (6) is μ^*TY _jo = 1, and meeting the optimal solution: μ^* > 0, ω^* > 0, so DMU _jo is called a weak DEA effective.

To solve the calculation and technical difficulties, using the duality theory to dual transformation the formula (6), and introduce the concept of non-Archimedes infinitesimal, then, it gets the C²R model of Non Archimedes infinitesimal. min [ θ j o - ɛ ( e ˆ T S - + e ˆ T S + ) ](7)s . t . { ∑ j = 1 n X j λ j + S − = Y j 0 ∑ j = 1 n Y j λ j − S + = Y j 0 λ j 0 ≥ 0 , j = 1 , 2 , ⋯ , n S − ≥ 0 S + ≥ 0(8)

In the formula (7), θ_j0 is the effectiveness of the decision-making unit of input-output, known as the coordinated development index; ɛ is Archimedes infinitesimal. Explain: any positive number is less than and greater than zero. S - = ( S 1 - , S 2 - , … , S m - ) T is a slack variable of m input. S + = ( S 1 + , S 2 + , … , S m + ) T is a slack variable of s output. λ = (λ₁, λ₂, …, λ _n ) is the n DMU combination coefficients; e ˆ = ( 1 , 1 , … , 1 ) ∈ E m , e = ( 1 , 1 , … , 1 ) T ∈ E s(9)

If the optimal solution of model is: λ * , S - * , S + * , θ j o * , according to the law of decision-making unit effectiveness: if θ j 0 * < 1 , DMU _jo is non DEA efficient. If θ j o * = 1 , e ˆ T S - * + e T S + * > 0 so DMU _jo is weak DEA efficient. If θ j o * = 1 , e ˆ T S - * + e T S + * = 0 , so DMU _jo is DEA efficient.

Malmquist productivity index is special index of measuring the total factor productivity change. Kraft (Caves) with others set up it on the basis of the Malmquist index and the concept of distance function. Because of the complexity of the calculation, the method is rarely used in practice. With the rapid development of DEA theory, scholars put them together, then Malmquist productivity index is widely used in the calculation of productivity.

This paper use Malmquist index based on DEA method to estimate the change of total factor productivity. And the Malmquist index method is built by Färe with others, and this method can examine the total factor productivity change (TFPCH) from t to (t+1).

If is D 0 t Based on the distance of the output function, and x is the input index and y is output index, so under the t period technology, from t to (t+1) Malmquist productivity index is: M t = D 0 t ( x t + 1 , y t + 1 ) D 0 t ( x t , y t )(10)

In the same way, under the (t+1) period technology, from t to (t+1) Malmquist productivity index is: M t + 1 = D 0 t + 1 ( x t + 1 , y t + 1 ) D 0 t + 1 ( x t , y t )(11)

In order to avoid error of an arbitrary choice, Färe define Malmquist index to the above two geometric average productivity index:

M 0 ( x t + 1 , y t + 1 , x t , y t ) = [ D 0 t + 1 ( x t + 1 , y t + 1 ) D 0 t + 1 ( x t , y t ) × D 0 t ( x t + 1 , y t + 1 ) D 0 t ( x t , y t ) ] 1 2(12)

Färe further pointed out that if the total factor productivity during t to (t + 1) is greater than 1, this indicates that the total factor productivity during the time is positive, it is growth period. Total factor productivity index could be decomposed into technical efficiency change (EFFCH) and technological progress (TECHCH), Which has the following relationship, total factor productivity change (TFPCH) = (EFFCH) technical efficiency change X technological progress (TECHCH), explain:

M 0 ( x t , y t , x t + 1 , y t + 1 ) = D 0 t + 1 ( x t + 1 , y t + 1 ) D 0 t ( x t , y t ) × [ D 0 t ( x t + 1 , y t + 1 ) D 0 t ( x t , y t ) × D 0 t + 1 ( x t + 1 , y t + 1 ) D 0 t + 1 ( x t , y t ) ] 1 2(13)

When pay the same scale, technical efficiency change (EFFCH) could be further decomposed into pure technical change components PEFFCH (based on the same scale reward) and scale efficiency change component SECH (based on the variable scale reward), the product is equal to the change in the technical efficiency (EFFCH), explain: EFFCH = PEFFCH × SECH.

The pure technical efficiency change (PEFFCH) is defined as: D 0 t ( x t + 1 , y t + 1 ) D 0 t ( x t , y t ) , Scale efficiency change (SECH) is defined as: S 0 t ( x t , y t ) S 0 t ( x t + 1 , y t + 1 ) , so the formula (11) could be represented as:

M 0 ( x t , y t , x t + 1 , y t + 1 ) = D 0 t ( x t + 1 , y t + 1 ) D 0 t ( x t , y t ) × S 0 t ( x t , y t ) S 0 t ( x t + 1 , y t + 1 ) × [ D 0 t ( x t + 1 , y t + 1 ) D 0 t ( x t , y t ) × D 0 t + 1 ( x t + 1 , y t + 1 ) D 0 t + 1 ( x t , y t ) ] 1 2(14)

According to the method proposed by Färe, the above formula can be calculated using the DEA linear programming method combined with panel data. SECH > 1, which means that the change of production system input concentration scale and scale efficiency has been improved; PEFFCH > 1, which means that the efficiency of technological innovation has been improved; TECHCH > 1, which means that production technology has been improved; TFPCH > 1, which means that overall productivity has been improved, and the evaluation object is in growth period and its production activities are meaningful. On the contrary, if the above indicators are less than 1, they indicate that the corresponding efficiency deteriorates.