Intelligent selection of strategic alliance partners in automobile manufacturing industry based on DEA and grey system theory

3.2.1 GM (1, 1) model

To obtain the grey predict model GM (1, 1), the original irregular data are first accumulated to generate relatively regular sequences which are then employed for modeling. After that, data obtained from the generated model undergo the inverse accumulated generating operation so as to work out the predicted values of original data [22].

The original data sequence is assumed as follows: X ( 0 ) = ( X ( 0 ) ( 1 ) , X ( 0 ) ( 2 ) , … , X ( 0 ) ( n ) )

Where X⁽⁰⁾ (k)≥0, k = 1, 2, …, n ; X⁽¹⁾ is 1-AGO Sequence of the original sequence X⁽⁰⁾: X ( 1 ) = ( X ( 1 ) ( 1 ) , X ( 1 ) ( 2 ) , … , X ( 1 ) ( n ) )

Where X ( 1 ) ( k ) = ∑ i = 1 k X ( 0 ) ( i ) , k = 1 , 2 , … , n ;

After the accumulation of original data, the data randomness is lowered. If the original sequence X⁽⁰⁾ and 1-AGO sequence X⁽¹⁾ meet the requirements of quasi-smoothness test:

ρ ( k ) = X ( 0 ) ( k ) X ( 1 ) ( k - 1 ) < 0.5 , quasi-exponent law test:

δ ( k ) = x ( 1 ) ( k ) x ( 1 ) ( k - 1 ) ∈ [ 1 , 1.5 ] , and class-compare verification: σ ( k ) = x ( 0 ) ( k ) x ( 0 ) ( k - 1 ) ∈ ( e - 2 n + 1 , e 2 n + 1 )

X⁽¹⁾ Sequence denotes the law of exponential growth; in other words, it can be represented by the linear first-order differential equation as below: dx ( 1 ) ( k ) dt + α X ( 1 ) ( k ) = μ(1)

In the equation, a is the development coefficient which reflects the developing trend of X⁽¹⁾ and original sequence X⁽⁰⁾; μ is the internal control grey number indicating changes in relation among data.

To work out α and μ, α ∧ = [ α , μ ] T is set as the vector to be estimated; since the analyzed sequence is discrete, through the discretization of dx ( 1 ) ( k ) dt in Formula (1) the following formula is obtained: dx ( 1 ) ( k ) dt = X ( 1 ) ( k ) - X ( 1 ) ( k - 1 ) , ( k = 2 , 3 , … , n )(2)

Let if: Z ( 1 ) ( k ) = μ X ( 1 ) ( k ) + ( 1 - μ ) X ( 1 ) ( k - 1 )(3)

In the formula (3), Z⁽¹⁾ (k) is known as the background value of formula (1), and the weighting coefficient μ ∈ [0, 1].

If μ is set as 0.5, then Z ( 1 ) ( k ) = 1 2(4)

At this point, through the discretization of Formula (1), the following is obtained: X ( 1 ) ( k ) - X ( 1 ) ( k - 1 ) + α Z ( 1 ) ( k ) = μ(5)

By adopting the least squares method to solve Formula (5), the following is obtained: α ∧ = ( B T B ) - 1 B T Y n(6)

Where B = [ - z ( 1 ) ( 2 ) 1 - z ( 1 ) ( 3 ) 1 ⋮ ⋮ - z ( 1 ) ( n ) 1 ] = [ - 1 2 [ x ( 1 ) ( 1 ) + x ( 1 ) ( 2 ) ] 1 - 1 2 [ x ( 1 ) ( 2 ) + x ( 1 ) ( 3 ) ] 1 ⋮ ⋮ - 1 2 [ x ( 1 ) ( n - 1 ) + x ( 1 ) ( n ) ] 1 ] , Y n = [ x ( 0 ) ( 2 ) x ( 0 ) ( 3 ) ⋮ x ( 0 ) ( n ) ]

After the calculation of α and μ, we continue to solve the Differential Equation (1) and the following formula is obtained [23]: X ( 1 ) ∧ = ce - α t + μ α(7)

In the formula, X ( 1 ) ∧ represents the predicted values of Sequence X⁽¹⁾, with c as the undefined constant.

The discretization of Formula (7) leads to Formula (8) as below: X ( 1 ) ∧ ( k + 1 ) = ce - α t + μ α ( k = 0 , 1 , … , n - 1 )(8)

To work out constant c, an initial value is determined beforehand, and it is assumed that X ( 1 ) ∧ ( 1 ) = X ( 0 ) ( 1 ) ; as a result, the following formula is obtained: c = X ( 0 ) ( 1 ) - μ α(9)

By substituting Formula (9) into Formula (8), the following is obtained: X ( 1 ) ∧ ( k + 1 ) = ( X ( 0 ) ( 1 ) - μ α ) e - α t + μ α(10)

Through the inverse accumulated generating operation of Formula (10), the grey prediction model of original sequence X⁽⁰⁾ is obtained as follows: X ( 0 ) ∧ ( k + 1 ) = ( X ( 0 ) ( 1 ) - μ α ) e - α t ( 1 - e α )(11)

The selection of strategic alliance partners is rather a complicated decision-making process. Traditional DEA models (CCR and BCC Model) are unable to well recognize decision-making units whose efficiency value is 1 [24], and the input factor is required to be non-radial [25]. Therefore, in this study, a non-radial super-efficiency SBM model is selected, so that the slackness of factor inputs and outputs can be fully taken into account in relevant analysis and evaluation.

If we assume the number of DMU is nand numbers of the input and output types for each DMU _j are respectively m and s, we obtain x _j = (x_1j, x_2j, …, x _mj ) ^T , y _j = (y_1j, y_2j, …, y _sj ) ^T in which χ _ij > 0 represents the input of the i ^th type for the j ^th DMU _j , and y _rj > 0 represents the output of the r ^th type for the j ^th DMU _j (j = 1, 2, …, n ; n ; i = 1, 2, …, m ; r = 1, 2, …, s). Si^- and S r + are respectively surplus and slack variables, λ _j is the weight vector, and χ₀, y₀ are respectively the input and output of the decision-making unit DMU₀. Therefore, to determine the efficiency of a selected DMU₀, the non-radial DEA model will be as follows: min ρ = 1 + 1 m ∑ i = 1 m s - i x i 0 1 1 s ∑ r - 1 s s r + y r 0(12)s . t ∑ j = 1 , ≠ 0 n λ j x j - S - ≤ x 0 , ∑ j = 1 , ≠ 0 n λ j x j + S + ≤ y 0(13)

eλ = 1, λ, S^-, S⁺ ≥ 0, ρ is the effective value of DMU₀. When ρ ≥ 1, DMU₀ is of relative efficiency in DEA; when ρ < 1, DMU₀ is not efficient in DEA, which means necessary upgrades should be made to inputs and outputs.

In order to fully measure the efficiency of the DEA model and meanwhile find the suitable alliance partner for the target decision-making unit, this paper, based on the input-output correlation, set fixed assets (Fix.as), costs of sales (Cogs), operating expenses (O.exp) and long-term investment (L.inv) as input factors, and revenue (Rev), total equity (T.eq) and net income (Net.in) as output factors. For proprietors and investors, these indicators can be employed to evaluate the operation and revenue of an enterprise, and on this basis, they can decide whether they will form alliance with the enterprise. The indicators are show in Table 2.