Consensus Modeling with Asymmetric Cost Based on Data-Driven Robust Optimization

Abstract

The robust optimization method has progressively become a research hot spot as a valuable means for dealing with parameter uncertainty in optimization problems. Based on the asymmetric cost consensus model, this paper considers the uncertainties of the experts’ unit adjustment costs under the background of group decision making. At the same time, four uncertain level parameters are introduced. For three types of minimum cost consensus models with direction restrictions, including MCCM-DC,$\varepsilon$-MCCM-DC and threshold-based (TB)-MCCM-DC, the robust cost consensus models corresponding to four types of uncertainty sets (Box set, Ellipsoid set, Polyhedron set and Interval-Polyhedron set) are established. Sensitivity analysis is carried out under different parameter conditions to determine the robustness of the solutions obtained from robust optimization models. The robust optimization models are then compared to the minimum cost models for consensus. The example results show that the Interval-Polyhedron set’s robust models have the smallest total costs and strongest robustness. Decision makers can choose the combination of uncertainty sets and uncertain levels according to their risk preferences to minimize the total cost. Finally, in order to reduce the conservatism of the classical robust optimization method, the pricing information of the new product MACUBE 550 is used to build a data-driven robust optimization model. Ellipsoid uncertainty set is proved to better trade-off the average performance and robust performance through different measurement indicators. Therefore, the uncertainty set can be selected according to the needs of the group.

Appendix

For $\varepsilon$-MCCM-DC, i.e., model (2.5), its robust minimum cost consensus problem is:

$$\begin{aligned} \begin{aligned}&\min \left\{ \underset{c_{i}^{U}\in {{{\mathcal {Z}}}^{U}},c_{i}^{D}\in {{{\mathcal {Z}}}^{D}}}{\mathop {\max }}\,\sum \limits _{i=1}^{m}{(c_{i}^{U}\varphi _{i}^{-}+c_{i}^{D}\varphi _{i}^{+})}{:}\,\right. \\&\left. \qquad {{o}^{c}}\in O,{{o}^{c}}+\varphi _{i}^{+}-\varphi _{i}^{-}={{o}_{i}}\text {,}\varphi _{i}^{+},\varphi _{i}^{-}\ge 0,\varphi _{i}^{+},\varphi _{i}^{-}\le {{\varepsilon }_{i}},i=1,2,\ldots ,m \right\} . \end{aligned} \end{aligned}$$

Then, its corresponding four uncertainty sets are as follows:

• Interval uncertainty

  $$\begin{aligned} \begin{aligned} \min&\quad \sum \limits _{i\in [m]}{(({\hat{c}}_{i}^{U}+\lambda ({\underline{c}}_{i}^{U}-{\hat{c}}_{i}^{U})})\varphi _{i}^{-}+({\hat{c}}_{i}^{D}+\lambda ({\underline{c}}_{i}^{D}-{\hat{c}}_{i}^{D}))\varphi _{i}^{+}) \\ s.t.&\quad {{o}^{c}}\in O \\&\quad {{o}^{c}}+\varphi _{i}^{+}-\varphi _{i}^{-}={{o}_{i}},i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-}\ge 0,i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-}\le {{\varepsilon }_{i}},i\in [m]. \\ \end{aligned} \end{aligned}$$

• Ellipsoidal uncertainty

  $$\begin{aligned} \begin{aligned} \min&\quad \sum \limits _{i\in [m]}{({\hat{c}}_{i}^{U}}\varphi _{i}^{-}+{\hat{c}}_{i}^{D}\varphi _{i}^{+})+u \\ s.t.&\quad \lambda ({{(\varphi _{i}^{-})}^{T}}\varSigma \varphi _{i}^{-}+{{(\varphi _{i}^{+})}^{T}}\varSigma \varphi _{i}^{+})\le {{u}^{2}},i\in [m] \\&\quad {{o}^{c}}\in O \\&\quad {{o}^{c}}+\varphi _{i}^{+}-\varphi _{i}^{-}={{o}_{i}},i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-}\ge 0,i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-}\le {{\varepsilon }_{i}},i\in [m]. \\ \end{aligned} \end{aligned}$$

• Polyhedral uncertainty

  $$\begin{aligned} \begin{aligned} \min&\quad u \\ s.t.&\quad \sum \limits _{i\in [m]}{\sum \limits _{j\in [M]}{(c_{ij}^{U}}\varphi _{i}^{-}+c_{ij}^{D}\varphi _{i}^{+})}\le u,i\in [m] \\&\quad {{o}^{c}}\in O \\&\quad {{o}^{c}}+\varphi _{i}^{+}-\varphi _{i}^{-}={{o}_{i}},i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-}\ge 0,i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-}\le {{\varepsilon }_{i}},i\in [m]. \\ \end{aligned} \end{aligned}$$

• Interval-Polyhedral uncertainty

  $$\begin{aligned} \begin{aligned} \min&\quad \sum \limits _{i\in [m]}{({\hat{c}}_{i}^{U}}\varphi _{i}^{-}+{\hat{c}}_{i}^{D}\varphi _{i}^{+})+\lambda u+{{\left\| v \right\| }_{1}} \\ s.t.&\quad (\bar{c}_{i}^{U}-{\hat{c}}_{i}^{U})\varphi _{i}^{-}+(\bar{c}_{i}^{D}-{\hat{c}}_{i}^{D})\varphi _{i}^{+}\le u+{{v}_{i}},i\in [m] \\&\quad {{o}^{c}}\in O \\&\quad {{o}^{c}}+\varphi _{i}^{+}-\varphi _{i}^{-}={{o}_{i}},i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-}\ge 0,i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-}\le {{\varepsilon }_{i}},i\in [m]. \\ \end{aligned} \end{aligned}$$

For TB-MCCM-DC, i.e., model (2.8), its robust minimum cost consensus problem is:

$$\begin{aligned} \begin{aligned}&\min \left\{ \underset{c_{i}^{U}\in {{{\mathcal {Z}}}^{U}},c_{i}^{D}\in {{{\mathcal {Z}}}^{D}}}{\mathop {\max }}\,\sum \limits _{i=1}^{m}{(c_{i}^{U}\varphi _{i}^{-}+c_{i}^{D}\varphi _{i}^{+})}{:}\, {{o}^{c}}\in O,\varphi _{i}^{+}-\varphi _{i}^{-}={{o}_{i}}-{{o}^{c}}+{{\eta }_{i}},\right. \\&\left. \qquad \phi _{i}^{+}-\phi _{i}^{-}={{o}_{i}}-{{o}^{c}}-{{\eta }_{i}},\varphi _{i}^{+},\varphi _{i}^{-},\phi _{i}^{+},\phi _{i}^{-}\ge 0,i=1,2,\ldots ,m \right\} . \end{aligned} \end{aligned}$$

Then, its corresponding four uncertainty sets are as follows:

• Interval uncertainty

  $$\begin{aligned} \begin{aligned} \min&\quad \sum \limits _{i\in [m]}{(({\hat{c}}_{i}^{U}+\lambda ({\underline{c}}_{i}^{U}-{\hat{c}}_{i}^{U})})\varphi _{i}^{-}+({\hat{c}}_{i}^{D}+\lambda ({\underline{c}}_{i}^{D}-{\hat{c}}_{i}^{D}))\varphi _{i}^{+}) \\ s.t.&\quad {{o}^{c}}\in O \\&\quad \varphi _{i}^{+}-\varphi _{i}^{-}={{o}_{i}}-{{o}^{c}}+{{\eta }_{i}},i\in [m] \\&\quad \phi _{i}^{+}-\phi _{i}^{-}={{o}_{i}}-{{o}^{c}}-{{\eta }_{i}},i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-},\phi _{i}^{+},\phi _{i}^{-}\ge 0,i\in [m].\\ \end{aligned} \end{aligned}$$

• Ellipsoidal uncertainty

  $$\begin{aligned} \begin{aligned} \min&\quad \sum \limits _{i\in [m]}{({\hat{c}}_{i}^{U}}\varphi _{i}^{-}+{\hat{c}}_{i}^{D}\varphi _{i}^{+})+u \\ s.t.&\quad \lambda ({{(\varphi _{i}^{-})}^{T}}\varSigma \varphi _{i}^{-}+{{(\varphi _{i}^{+})}^{T}}\varSigma \varphi _{i}^{+})\le {{u}^{2}},i\in [m] \\&\quad {{o}^{c}}\in O \\&\quad \varphi _{i}^{+}-\varphi _{i}^{-}={{o}_{i}}-{{o}^{c}}+{{\eta }_{i}},i\in [m] \\&\quad \phi _{i}^{+}-\phi _{i}^{-}={{o}_{i}}-{{o}^{c}}-{{\eta }_{i}},i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-},\phi _{i}^{+},\phi _{i}^{-}\ge 0,i\in [m]. \\ \end{aligned} \end{aligned}$$

• Polyhedral uncertainty

  $$\begin{aligned} \begin{aligned} \min&\quad u \\ s.t.&\quad \sum \limits _{i\in [m]}{\sum \limits _{j\in [M]}{(c_{ij}^{U}}\varphi _{i}^{-}+c_{ij}^{D}\varphi _{i}^{+})}\le u,i\in [m] \\&\quad {{o}^{c}}\in O \\&\quad \varphi _{i}^{+}-\varphi _{i}^{-}={{o}_{i}}-{{o}^{c}}+{{\eta }_{i}},i\in [m] \\&\quad \phi _{i}^{+}-\phi _{i}^{-}={{o}_{i}}-{{o}^{c}}-{{\eta }_{i}},i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-},\phi _{i}^{+},\phi _{i}^{-}\ge 0,i\in [m] .\\ \end{aligned} \end{aligned}$$

• Interval-Polyhedral uncertainty

  $$\begin{aligned} \begin{aligned} \min&\quad \sum \limits _{i\in [m]}{({\hat{c}}_{i}^{U}}\varphi _{i}^{-}+{\hat{c}}_{i}^{D}\varphi _{i}^{+})+\lambda u+{{\left\| v \right\| }_{1}} \\ s.t.&\quad (\bar{c}_{i}^{U}-{\hat{c}}_{i}^{U})\varphi _{i}^{-}+(\bar{c}_{i}^{D}-{\hat{c}}_{i}^{D})\varphi _{i}^{+}\le u+{{v}_{i}},i\in [m] \\&\quad {{o}^{c}}\in O \\&\quad \varphi _{i}^{+}-\varphi _{i}^{-}={{o}_{i}}-{{o}^{c}}+{{\eta }_{i}},i\in [m] \\&\quad \phi _{i}^{+}-\phi _{i}^{-}={{o}_{i}}-{{o}^{c}}-{{\eta }_{i}},i\in [m] \\&\quad \varphi _{i}^{+},\varphi _{i}^{-},\phi _{i}^{+},\phi _{i}^{-}\ge 0,i\in [m] .\\ \end{aligned} \end{aligned}$$