Multi-attribute decision-making methods based on normal random variables in supply chain risk management

After studying various methods of determining objective weights, some scholars have proposed mathematical optimisation models [4]. These models often use the following methods when solving objective weights. We transform the exact number decision matrix A = (a_ij)_m×n into a standardised decision matrix B = (b_ij)_m×n. (6) bij=aij−ajminajmax−ajmin, j∈J1; {b_{ij}} = {{{a_{ij}} - a_j^{\min}} \over {a_j^{\max} - a_j^{\min}}},\quad j \in {J_1}; (7) bij=ajmax−aijajmax−ajmin, j∈J2; {b_{ij}} = {{a_j^{\max} - {a_{ij}}} \over {a_j^{\max} - a_j^{\min}}},\quad j \in {J_2};

J₁ is a profit-based indicator. J₂ is a cost index. In this way, a solution model for objective weights is obtained (8) (8) {minZ1=∑j=1nwTHws.t.eTw=1, wj≥0 \left\{{\matrix{{\min {Z_1} = \sum\limits_{j = 1}^n {w^T}Hw} \hfill \cr {s.t.{e^T}w = 1,\quad {w_j} \ge 0} \hfill \cr}} \right.

H is the diagonal matrix of n × n. Its diagonal element is hij=∑i=1m(bij−bj*)2 {h_{ij}} = \sum\limits_{i = 1}^m {\left({{b_{ij}} - {b_j}^*} \right)^2} , b_j^* = max {b_1j ⋯ b_mj}. Solving model (8) can get: (9) w=H−1e/eTH−1e w = {H^{- 1}}e/{e^T}{H^{- 1}}e

Some scholars pointed out that the weight distribution mechanism as well as the meaning of model (8) is not precise, and it does not conform to the principle of entropy model weight distribution. Through case analysis, it is found that small changes in the decision matrix will lead to significant changes in weights, so the weight distribution mechanism of the model (8) is unreasonable. So, we proposed an entropy model to solve the objective weights [5]. The main methods are as follows: (10) wj=dj/∑j=1ndjdj=1−Ej, Ej=−(∑i=1npijlnpij)/lnnpij=aij/∑i=1maij \matrix{{{w_j} = {d_j}/\sum\limits_{j = 1}^n {d_j}} \hfill \cr {{d_j} = 1 - {E_j},\quad {E_j} = - \left({\sum\limits_{i = 1}^n {p_{ij}}\ln {p_{ij}}} \right)/\ln n{p_{ij}} = {a_{ij}}/\sum\limits_{i = 1}^m {a_{ij}}} \hfill \cr}

To assign weights, the entropy model is guided by the following principle. If the evaluation value of each scheme under the j attribute tends to be more consistent, then the weight of the j attribute will be smaller. The entropy model also has some unreasonable points in assigning weights as follows:

The weight distribution is not flexible. The entropy model defines d_j = 1 − E_j, so can d_j = 2 − E_j or other functions of E_j being set?

The construction of a suitable mathematical model requires a deep understanding of the specific situation and rich mathematical experience of the issues involved in the decision-making problem. This isn’t easy. To judge the rationality of the objective weight model, we give Judgement Theorem 1.

Based on the model (8) and entropy model (10), we transform the exact number decision matrix A = (a_ij)_m×n into a standardised decision matrix C = (c_ij)_m×n. Among them: (11) cij=aijajmax, j∈J1; {c_{ij}} = {{{a_{ij}}} \over {a_j^{\max}}},\quad j \in {J_1}; (12) cij=ajminaij, j∈J2; {c_{ij}} = {{a_j^{\min}} \over {{a_{ij}}}},\quad j \in {J_2};

J₁ is a profit-based indicator and J₂ is a cost index; (13) ajmax=max{a1j,a2j,⋯,amj}, j=1,⋯,n;ajmin=min{a1j,a2j,⋯,amj}, j=1,⋯,n; \matrix{{a_j^{\max} = \max \left\{{{a_{1j}},{a_{2j}}, \cdots ,{a_{mj}}} \right\},\quad j = 1, \cdots ,n;} \hfill \cr {a_j^{\min} = \min \left\{{{a_{1j}},{a_{2j}}, \cdots ,{a_{mj}}} \right\},\quad j = 1, \cdots ,n;} \hfill \cr}

For the normalised matrix C = (c_ij)_m×n, the entropy of the j attribute is defined as: (14) hj=ρ−Ej {h_j} = \rho - {E_j}

Ej=−(∑i=1ncijlncij)/lnn {E_j} = - \left({\sum\limits_{i = 1}^n {c_{ij}}\ln {c_{ij}}} \right)/\ln n , ρ is the system parameter (ρ ≥ max{E₁,⋯E_j ⋯E_n}). Then the entropy coefficient model for solving the objective weight is: (15) {minZ2=wTKws.t.eTw=1w≥0 \left\{{\matrix{{\min {Z_2} = {w^T}Kw} \hfill \cr {s.t.{e^T}w = 1} \hfill \cr {w \ge 0} \hfill \cr}} \right. where K is the diagonal matrix of n×n. Its diagonal elements are k_ij = ρ −E_j, k_ij > 0, j = 1,⋯ ,n; the remaining elements are zero. We assume that L = w^T Kw − λ(e^T w − 1), then ∂L∂wj=2Kw−λ=0 {{\partial L} \over {\partial {w_j}}} = 2Kw - \lambda = 0 , ∂L∂λ=eTw−1=0 {{\partial L} \over {\partial \lambda}} = {e^T}w - 1 = 0 . Calculate to get w = K⁻¹e/e^T K⁻¹e.

Property 1. The weight distribution principle of the entropy coefficient model is the same as that of the entropy model. If the evaluation value of each scheme under the jth attribute tends to be more consistent, then the weight of the jth attribute will be smaller.

Property 2. The entropy coefficient model has certain flexibility [7]. The decision-maker can set the size of the system parameter ρ according to specific actual needs to adjust the degree of the weight difference between attributes. The larger the ρ, the smaller the system attribute weight difference.

Here are two examples to illustrate the difference between the entropy coefficient model (14), the model (8), and the entropy model (10):

Suppose there is a decision matrix Am×n=[a11⋯a1n⋮⋮⋮am1⋯amn] {A_{m \times n}} = \left[ {\matrix{{{a_{11}}} & \cdots & {{a_{1n}}} \cr \vdots & \vdots & \vdots \cr {{a_{m1}}} & \cdots & {{a_{mn}}} \cr}} \right] . We use model (8) to normalise A to get matrix B, then wj=(hjj)−1∑j=1n(hjj)−1 {w_j} = {{{{\left({{h_{jj}}} \right)}^{- 1}}} \over {\sum\limits_{j = 1}^n {{\left({{h_{jj}}} \right)}^{- 1}}}} can be obtained according to formula (9). If the evaluation value of each scheme under the j attribute tends to be the same [8]. That is, b_ij → b_j* (i = 1, 2,⋯ ,m) is h_ij → 0, so wj=limh→0jj(hjj)−1∑j=1n(hjj)−1=1 {w_j} = \mathop {\lim}\limits_{\mathop {h \to 0}\limits_{jj}} {{{{\left({{h_{jj}}} \right)}^{- 1}}} \over {\sum\limits_{j = 1}^n {{\left({{h_{jj}}} \right)}^{- 1}}}} = 1 . Model (8) may cause the weight of the j attribute to be too large.

We use the entropy model (10) to calculate. If the evaluation value of each scheme under the jth attribute tends to be the same, that is, p_ij → 1/n(i = 1,2⋯m) is d_j → 0, so wj=dj∑j=1ndj→0 {w_j} = {{{d_j}} \over {\sum\limits_{j = 1}^n {d_j}}} \to 0 . When assigning weights, the degree weight difference may be too large.

3. We use the entropy coefficient model (15) for solving objective weight. According to formula (15), we can get: wj=(kjj)−1∑j=1n(kjj)−1 {w_j} = {{{{\left({{k_{jj}}} \right)}^{- 1}}} \over {\sum\limits_{j = 1}^n {{\left({{k_{jj}}} \right)}^{- 1}}}} . If the evaluation value of each scheme under the jth attribute tends to be the same, that is, c_ij → 1(i = 1,2,⋯ ,m) is E_j → 0. So k_jj = ρ − E_j → ρ, then (16) wj=limk→ρjj(kjj)−1∑j=1n(kjj)−1=ρ−1∑i≠jn(kjj)−1+ρ−1 {w_j} = \mathop {\lim}\limits_{\mathop {k \to \rho}\limits_{jj}} {{{{\left({{k_{jj}}} \right)}^{- 1}}} \over {\sum\limits_{j = 1}^n {{\left({{k_{jj}}} \right)}^{- 1}}}} = {{{\rho ^{- 1}}} \over {\sum\limits_{i \ne j}^n {{\left({{k_{jj}}} \right)}^{- 1}} + {\rho ^{- 1}}}}

In this way, we can set the system parameters ρ according to the specific decision-making situation so that the entropy coefficient model (15) has a certain degree of flexibility [9].

There is a decision matrix A_4×4. To simplify, we assume that its attribute indicators are all income indicators (17) A=[P1P2P3P4S130303829.0S219548629.0S319158528.9S468706029.0] A = \left[ {\matrix{{} & {{P_1}} & {{P_2}} & {{P_3}} & {{P_4}} \cr {{S_1}} & {30} & {30} & {38} & {29.0} \cr {{S_2}} & {19} & {54} & {86} & {29.0} \cr {{S_3}} & {19} & {15} & {85} & {28.9} \cr {{S_4}} & {68} & {70} & {60} & {29.0} \cr}} \right]

Using model (8), we can get: w = (0.1384, 0.2232, 0.2783, 0.3601).

When a₃₄ undergoing a small change, the weight change obtained using model (8) is too large [10]. The weight of the particular attribute P4 has changed from 0.3601 to 0.1579. The weights obtained by using the entropy model (10) have not changed. This cannot reflect a slight change in the decision matrix. Using the entropy coefficient model (14), the weight change obtained is relatively small, consistent with the slight chance of the matrix. From the aforementioned two examples, the entropy coefficient model can adapt to different decision-making situations by adjusting the value of the system parameter ρ, and the weight obtained is more reasonable than the model (8) and the entropy model (10).

Suppose that the decision-maker directly gives the emotional weight of the attribute as W(0)=(w1(0),⋯,wj(0),⋯wn(0)) {W^{(0)}} = \left({w_1^{(0)}, \cdots ,w_j^{(0)}} \right.,\left. {\cdots w_n^{(0)}} \right) , 0≤wj(0)≤1 0 \le w_j^{(0)} \le 1 and ∑j=1nwj(0)=1 \sum\limits_{j = 1}^n w_j^{(0)} = 1 . The total weight of the attribute: (19) W*=(w1*,⋯,wj*,⋯wn*)wj*=βwj(0)+(1−β)wj∑j=1nwj*=β∑j=1nwj(0)+(1−β)∑j=1nwj=1 \matrix{{{W^*} = \left({{w_1}^*, \cdots ,{w_j}^*, \cdots {w_n}^*} \right)} \hfill \cr {{w_j}^* = \beta w_j^{(0)} + \left({1 - \beta} \right){w_j}\sum\limits_{j = 1}^n {w_j}^* = \beta \sum\limits_{j = 1}^n w_j^{(0)} + \left({1 - \beta} \right)\sum\limits_{j = 1}^n {w_j} = 1} \hfill \cr}

β (0 ≤ β ≤ 1) is the weighted trade-off coefficient. If the ranking of the schemes is highly sensitive to weight changes, the reliability of the evaluation results is difficult to guarantee. It is also tricky for decision-makers to make choices [11]. To judge the rationality of the total weights, Judgement Theorem 2 is proposed.