A strategic scheme for partnership supply networks focusing on green multi-agent transportations: a game theory approach

Abstract

Competition to gain more market share leads to the development of a dynamic environment and the fulfillment of an improvement cycle in supply networks. In this paper, as the first attempt, two strategic periods based on the introduction and growth phases of supply networks are considered. Each of these periods has its own characteristics and develops different competitive structures. The allocation of market share between competing supply networks is carried out on the extent of their relationships with green orientations in the network. The ownership of supply sources in these networks is replaced by that of relationships (partnership supply) developed by some parent brands. The efforts of each parent brand towards establishing green supply relationships lead to a greater market share for it. Further to competition between the supply networks, co-petition within each network also results in the emergence of different virtual alliance structures. The performance history of each brand and its subsidiary partners in the first period is included in the second period’s assessments and leads to the development or decline of the network. Another dimension of the paper lies in the development of multi-agent transportation arising from the green requirements. Two gaming-based heuristic algorithms are proposed for handling the decentralized decision-making structures in the multi-period multi-stage platform under discussion. To evaluate the taken approach, a real-world inspired case is taken into account. The numerical results prove the better performance of the proposed scheme towards green coverage of the network and, thereupon, increased market share.

Appendices

Appendix 1

$${Obj}_{\begin{array}{c}{r}_g{r}_g^{\prime}\\ {}\ \end{array}}^{gg}=\mathit{\max}\kern0.5em \left[{\lambda}_g\sum_{\begin{array}{c}{r}_g^{\prime \prime}\ne {r}_g,{r}_g^{\prime}\\ {}\ \end{array}}^{R_g}{IG_g}_{r_g^{\prime \prime }t}^{r_g}+{\lambda}_{or}\sum_{\begin{array}{c}{r}_{or}=1\\ {}\ \end{array}}^{R_{or}}{IG_{or}}_{r_{or}t}^{r_g}\right]$$

(1)

$${\displaystyle \begin{array}{cc}{Obj}_{\begin{array}{c}{r}_g{r}_{or}\\ {}\ \end{array}}^{gor}=\mathit{\max}& \left[\Big[{\lambda}_g\sum_{\begin{array}{c}{r}_g^{\prime}\ne {r}_g\\ {}\ \end{array}}^{R_g}{IG_g}_{r_g^{\prime }t}^{r_g}+{\lambda}_{or}\sum_{\begin{array}{c}{r}_{or}^{\prime}\ne {r}_{or}\\ {}\ \end{array}}^{R_{or}}{IG_{or}}_{r_{or}^{\prime }t}^{r_g}\right]\end{array}}$$

(2)

$${\displaystyle \begin{array}{cc}{Obj}_1^i=\mathit{\max}& \left[\Big[{\lambda}_g\sum_{\begin{array}{c}{r}_g=1\\ {}\ \end{array}}^{R_g}{G_g}_{r_gt}^i+{\lambda}_{or}\sum_{\begin{array}{c}{r}_{or}=1\\ {}\ \end{array}}^{R_{or}}{G_{or}}_{r_{or}t}^i\right]\end{array}}$$

(3)

$${\displaystyle \begin{array}{l}{Obj}_2^i=\max \kern1.5em \left[{\lambda}_g\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_g=1\\ {}\ \end{array}}^{J_g}{Sp_g}_{pt}^i{Nu_g}_{j_g pvt}^i+{\lambda}_{or}\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_{or}=1\\ {}\ \end{array}}^{J_{or}}{Sp_{or}}_{pt}^i{Nu_{or}}_{j_{or} pvt}^i\right.\kern9.25em \\ {}\kern7.75em -{Cn}^i\left(\sum_{\begin{array}{c}{r}_g=1\\ {}\ \end{array}}^{R_g}{G_g}_{r_gt}^i+\sum_{\begin{array}{c}{r}_{or}=1\\ {}\ \end{array}}^{R_{or}}{G_{or}}_{r_{or}t}^i\right)-\sum_{p=1}^P\sum_{\begin{array}{c}{j}_g=1\\ {}\ \end{array}}^{J_g}{Fb_g}_{j_g pt}^i{Z_g}_{j_gt}^i\\ {}\begin{array}{l}\kern7.75em -\sum_{p=1}^P\sum_{\begin{array}{c}{j}_{or}=1\\ {}\ \end{array}}^{J_{or}}{Fb_{or}}_{j_{or} pt}^i{Z_{or}}_{j_{or}t}^i-\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_g=1\\ {}\ \end{array}}^{J_g}{Bu_g}_{j_g pt}^i{Nu_g}_{j_g pvt}^i\\ {}\left.\kern7.75em -\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_{or}=1\\ {}\ \end{array}}^{J_{or}}{Bu_{or}}_{j_{or} pt}^i{Nu_{or}}_{j_{or} pvt}^i\right]\end{array}\end{array}}$$

(4)

$${\displaystyle \begin{array}{l}{Obj}_1=\mathit{\min}\kern1.25em \left[\sum_{i=1}^I\sum_{p=1}^P\sum_{v=1}^v\sum_{\begin{array}{c}{j}_g=1\\ {}\ \end{array}}^{J_g}{Up_g}_{j_g pt}{Nu_g}_{j_g pvt}^i+\sum_{i=1}^I\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_{or}=1\\ {}\ \end{array}}^{J_{or}}{Up_{or}}_{j_{or} pt}{Nu_{or}}_{j_{or} pvt}^i\right.\kern9.25em \\ {}\kern7.5em +\mathrm{T}c\left(\sum_{\begin{array}{c}j\in {\Delta }_g\cup {\Delta }_{or}\\ {}\ \end{array}}\sum_{v\in V}{d}_{0j}\left({Q}_{0 jvt}+{Q}_{j0 vt}^{\prime}\right)+\sum_{\begin{array}{c}j,{j}^{\prime}\in {\Delta }_g\cup {\Delta }_{or}\\ {}\ \end{array}}\sum_{v\in V}{d}_{j{j}^{\prime }}{Q}_{j{j}^{\prime } vt}\right)\\ {}\begin{array}{l}\kern7.5em +{Pun}_{or}\left(\sum_{\begin{array}{c}j\in {\Delta }_{or}\\ {}\ \end{array}}\sum_{v\in V}{Q}_{0 jvt}+\sum_{\begin{array}{c}{j}^{\prime}\in {\Delta }_g\\ {}\ \end{array}}\sum_{\begin{array}{c}j\in {\Delta }_{or}\\ {}\ \end{array}}\sum_{v\in V}{Q}_{j^{\prime } jvt}+\sum_{\begin{array}{c}{j}^{\prime },j\in {\Delta }_{or},{j}^{\prime}\ne j\\ {}\ \end{array}}\sum_{v\in V}{Q}_{j^{\prime } jvt}\right)\\ {}\left.\kern7.5em +\sum_{i=1}^I\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_g=1\\ {}\ \end{array}}^{J_g}{Pro}_{pt}^g{Nu_g}_{j_g pvt}^i+\sum_{i=1}^I\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_{or}=1\\ {}\ \end{array}}^{J_{or}}{Pro}_{pt}^{or}{Nu_{or}}_{j_{or} pvt}^i\right]\end{array}\end{array}}$$

(5)

$${\displaystyle \begin{array}{l}{Obj}_2=\mathit{\min}\kern2em \left[\sum_{i=1}^I\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{R_g^i}\sum_{p=1}^P{Dn_g}_{p{\overline{r}}_g^it}\left({{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^i vt}^i+{{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^iv\mathrm{t}}^i\right)\right.\kern9.25em \\ {}\kern8em +\sum_{i=1}^I\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}\sum_{p=1}^P{Dn_{or}}_{p{\overline{r}}_{or}^it}\left({{Nu^{\prime}}_{or}^g}_{p{\overline{r}}_{or}^i vt}^i+{{Nu^{\prime}}_{or}^{or}}_{p{\overline{r}}_{or}^i vt}^i\right)\\ {}\begin{array}{l}\kern8em + Tc\left(\sum_{\begin{array}{c}\overline{r}\in \\ {}\sum_i{\varphi}_g^i\cup \sum_i{\varphi}_{or}^i\\ {}\ \end{array}}\sum_{v\in V}{d}_{0\overline{r}}^{\prime}\left({U}_{0\overline{r} vt}+{U}_{\overline{r}0 vt}^{\prime}\right)+\sum_{\begin{array}{c}\overline{r},{\overline{r}}^{\prime}\in \\ {}\sum_i{\varphi}_g^i\cup \sum_i{\varphi}_{or}^i\\ {}\ \\ {}\ \end{array}}\sum_{v\in V}{d}_{\overline{r}{\overline{r}}^{\prime}}^{\prime }{U}_{\overline{r}{\overline{r}}^{\prime } vt}\right)\\ {}\left.\kern8.25em +{Pun}_{or}^{\prime}\left(\sum_{\overline{r}\in \sum_i{\varphi}_{or}^i}\sum_{v\in V}{U}_{0\overline{r} vt}+\sum_{{\overline{r}}^{\prime}\in \sum_i{\varphi}_g^i}\sum_{\overline{r}\in \sum_i{\varphi}_{or}^i}\sum_{v\in V}{U}_{{\overline{r}}^{\prime}\overline{r} vt}+\sum_{\begin{array}{c}\begin{array}{c}{\overline{r}}^{\prime },\overline{r}\in \sum_i{\varphi}_{or}^i,\\ {}{\overline{r}}^{\prime}\ne \overline{r}\end{array}\\ {}\ \end{array}}\sum_{v\in V}{U}_{{\overline{r}}^{\prime}\overline{r} vt}\right)\right]\end{array}\end{array}}$$

(6)

$${\displaystyle \begin{array}{l}{Obj}_3^i=\mathit{\min}\kern2em \left[\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{As_g^g}_{p{\overline{r}}_g^it}^i{{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^i vt}^i+\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{As_{or}^g}_{p{\overline{r}}_{or}^it}^i{{Nu^{\prime}}_{or}^g}_{p{\overline{r}}_{or}^i vt}^i\right.\kern9.25em \\ {}\kern8em +\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{As_g^{or}}_{p{\overline{r}}_g^it}^i{{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^it}^i+\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{As_{or}^{or}}_{p{\overline{r}}_{or}^it}^i{{Nu^{\prime}}_{or}^{or}}_{p{\overline{r}}_{or}^it}^i\\ {}\begin{array}{l}\kern7.75em +\sum_{p=1}^P\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{Co_g^g}_{p{\overline{r}}_g^it}^i{{Nu^{\prime \prime}}_g^g}_{p{\overline{r}}_g^it}^i+\sum_{p=1}^P\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{{C\mathrm{o}}_{or}^g}_{p{\overline{r}}_{or}^it}^i{{Nu^{\prime \prime}}_{or}^g}_{p{\overline{r}}_{or}^it}^i\\ {}\begin{array}{c}\begin{array}{c}\kern7.75em +\sum_{p=1}^P\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{Co_g^{or}}_{p{\overline{r}}_g^it}^i{{Nu^{\prime \prime}}_g^{or}}_{p{\overline{r}}_g^it}^i+\sum_{p=1}^P\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{Co_{or}^{or}}_{p{\overline{r}}_{or}^it}^i{{Nu^{\prime \prime}}_{or}^{or}}_{p{\overline{r}}_{or}^it}^i\\ {}-\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{Pu_g^g}_{p{\overline{r}}_g^it}^i\left({{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^i vt}^i-{{Nu^{\prime \prime}}_g^g}_{p{\overline{r}}_g^it}^i\right)\\ {}\kern2.25em -\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{Pu_{or}^g}_{p{\overline{r}}_{or}^it}^i\left({{Nu^{\prime}}_{or}^g}_{p{\overline{r}}_{or}^i vt}^i-{{Nu^{\prime \prime}}_{or}^g}_{p{\overline{r}}_{or}^it}^i\right)\end{array}\\ {}\kern0.5em -\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{Pu_g^{or}}_{p{\overline{r}}_g^it}^i\left({{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^i vt}^i-{{Nu^{\prime \prime}}_g^{or}}_{p{\overline{r}}_g^it}^i\right)\\ {}\kern2em -\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{Pu_{or}^{or}}_{p{\overline{r}}_{or}^it}^i\left({{Nu^{\prime}}_{or}^{or}}_{p{\overline{r}}_{or}^i vt}^i-{{Nu^{\prime \prime}}_{or}^{or}}_{p{\overline{r}}_{or}^it}^i\right)\end{array}\end{array}\end{array}}$$

(7)

$${\displaystyle \begin{array}{cc}{Obj_g^1}_{\begin{array}{c}{\overline{r}}_g^i\\ {}\ \end{array}}^i=\mathit{\max}& \left[{\alpha}_g^g\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Dem_g^g}_{p{\overline{r}}_g^it}^i+{\alpha}_g^{or}\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Dem_g^{or}}_{p{\overline{r}}_g^it}^i\right]\end{array}}$$

(8)

$${\displaystyle \begin{array}{l}{Obj_g^2}_{\begin{array}{c}{\overline{r}}_g^i\\ {}\ \end{array}}^i=\mathit{\max}\kern2em \left[{\lambda}_g\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Se_g}_{pt}^i{{Nu^{\prime \prime}}_g^g}_{p{\overline{r}}_g^it}^i+{\lambda}_{or}\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Se_{or}}_{pt}^i{{Nu^{\prime \prime}}_g^{or}}_{p{\overline{r}}_g^it}^i\right.\kern9.25em \\ {}\kern9.25em -\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P\sum_{v=1}^V{Sp_g}_{pt}^i{{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^i vt}^i-\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P\sum_{v=1}^V{Sp_{or}}_{pt}^i{{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^i vt}^i\\ {}\begin{array}{l}\kern9.25em -\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Ac_g^g}_{p{\overline{r}}_g^it}^i{Dem_g^g}_{p{\overline{r}}_g^it}^i-\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Ac_g^{or}}_{p{\overline{r}}_g^it}^i{Dem_g^{or}}_{p{\overline{r}}_g^it}^i\kern5.75em \\ {}\kern9.25em \begin{array}{l}-\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P\sum_{v=1}^V{Hc_g^g}_{p{\overline{r}}_g^it}^i\left(\frac{{{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^i vt}^i}{2}\right)-\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P\sum_{v=1}^V{Hc_g^{or}}_{p{\overline{r}}_g^it}^i\left(\frac{{{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^i vt}^i}{2}\right)\\ {}\begin{array}{l}-\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P\sum_{v=1}^V{Pu_g^g}_{p{\overline{r}}_g^it}^i\left({{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^i vt}^i-{{Nu^{\prime \prime}}_g^g}_{p{\overline{r}}_g^it}^i\right)\\ {}\left.-\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P\sum_{v=1}^V{Pu_g^{or}}_{p{\overline{r}}_g^it}^i\left({{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^i vt}^i-{{Nu^{\prime \prime}}_g^{or}}_{p{\overline{r}}_g^it}^i\right)\right]\end{array}\end{array}\end{array}\end{array}}$$

(9)

Appendix 2

$${\displaystyle \begin{array}{c}{Obj}_{r_g}^g=\max \kern3.75em \left[\sum_{i=1}^I\left({\lambda}_g\right.\left(\frac{\sum_{\begin{array}{c}{\overline{r}}_g^i\\ {}\ \end{array}}^{{\overline{R}}_g^i}\sum_p^P{{Nu^{\prime \prime}}_g^g}_{p{\overline{r}}_g^it-1}^i+\sum_{\begin{array}{c}{\overline{r}}_{or}^i\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}\sum_p^P{{Nu^{\prime \prime}}_{or}^g}_{p{\overline{r}}_{or}^it-1}^i}{\sum_{j_g}^{J_g}\sum_p^P{Nu_g}_{j_g pt-1}^i}\right)\right.\kern4em \\ {}\left.\kern9.75em +{\lambda}_{or}\left(\frac{\sum_{\begin{array}{c}{\overline{r}}_g^i\\ {}\ \end{array}}^{{\overline{R}}_g^i}\sum_p^P{{Nu^{\prime \prime}}_g^{or}}_{p{\overline{r}}_g^it-1}^i+\sum_{\begin{array}{c}{\overline{r}}_{or}^i\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}\sum_p^P{{Nu^{\prime \prime}}_{or}^{or}}_{p{\overline{r}}_{or}^it-1}^i}{\sum_{j_{\begin{array}{c} or\\ {}\ \end{array}}}^{J_{or}}\sum_p^P{Nu_{or}}_{\begin{array}{c}{j}_{or} pt-1\\ {}\ \end{array}}^i}\right)\left({De_{\mathrm{g}}}_{r_g}^i\right)\right]\end{array}}$$

(10)

$${\displaystyle \begin{array}{c}{Obj}_{{\overline{r}}_g^i}^{\overline{g}}=\mathit{\max}\kern3.75em \left[\sum_{i=1}^I\left({\lambda}_g\right.\left(\frac{\sum_{\begin{array}{c}{{\overline{r}}^{\prime}}_g^i\ne {\overline{r}}_g^i\\ {}\ \end{array}}^{{\overline{R}}_g^i}\sum_p^P{{Nu^{\prime \prime}}_g^g}_{\begin{array}{c}p{{\overline{r}}^{\prime}}_g^it-1\\ {}\ \end{array}}^i+\sum_{\begin{array}{c}{\overline{r}}_{or}^i\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}\sum_p^P{{Nu^{\prime \prime}}_{or}^g}_{\begin{array}{c}p{\overline{r}}_{or}^it-1\\ {}\ \end{array}}^i}{\sum_{\begin{array}{c}{j}_g\\ {}\ \end{array}}^{J_g}\sum_p^P{Nu_g}_{\begin{array}{c}{j}_g pt-1\\ {}\ \end{array}}^i}\right)\right.\kern4em \\ {}\left.\kern10.25em +{\lambda}_{or}\left.\left(\frac{\sum_{\begin{array}{c}{{\overline{r}}^{\prime}}_g^i\ne {\overline{r}}_g^i\\ {}\ \end{array}}^{{\overline{R}}_g}\sum_p^P{{Nu^{\prime \prime}}_g^{or}}_{\begin{array}{c}p{{\overline{r}}^{\prime}}_g^it-1\\ {}\ \end{array}}^i+\sum_{{\overline{r}}_{or}}^{{\overline{R}}_{or}}\sum_p^P{{Nu^{\prime \prime}}_{or}^{or}}_{\begin{array}{c}p{\overline{r}}_{or}t-1\\ {}\ \end{array}}^i}{\sum_{\begin{array}{c}{j}_{or}\\ {}\ \end{array}}^{J_{or}}\sum_p^P{Nu_{or}}_{\begin{array}{c}{j}_{or} pt-1\\ {}\ \end{array}}^i}\right)\right)\left({De_g}_{r_g}^i\right)\right]\end{array}}$$

(11)

$${Obj}_1^i=\mathit{\max}\kern3.5em \left[{\lambda}_g\sum_{\begin{array}{c}{j}_g=1\\ {}\ \end{array}}^{J_g}{Z_g}_{j_g}^i+{\lambda}_{or}\sum_{\begin{array}{c}{j}_{or}=1\\ {}\ \end{array}}^{J_{or}}{Z_{or}}_{j_{or}}^i\right]$$

(12)

$${\displaystyle \begin{array}{l}{Obj}_1^i=\mathit{\max}\kern3.5em \left[{\lambda}_g\left(\sum_{\begin{array}{c}{r}_g=1\\ {}\ \end{array}}^{R_g}{G_g}_{r_g}^i+\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g}{{\overline{G}}_g}_{{\overline{r}}_g^i}^i+\sum_{i^{\prime}\ne i}^I\sum_{\begin{array}{c}{\overline{r}}_g^{i^{\prime }}\\ {}\ \end{array}}^{{\overline{R}}_g^{i^{\prime }}}{{\overline{\overline{G}}}_g}_{{\overline{r}}_g^{i^{\prime}}}^i\right)+{\lambda}_{or}\left(\sum_{\begin{array}{c}{r}_{or}=1\\ {}\ \end{array}}^{R_{or}}{G_{or}}_{r_{or}}^i\right.\right]\kern3.75em \\ {}\left.\left.\kern9.75em +\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}}{{\overline{G}}_{or}}_{{\overline{r}}_{or}^i}^i+\sum_{i^{\prime}\ne i}^I\sum_{\begin{array}{c}{\overline{r}}_{o\mathrm{r}}^{i^{\prime }}\\ {}\ \end{array}}^{{\overline{R}}_{or}^{i^{\prime }}}{{\overline{\overline{G}}}_{or}}_{{\overline{r}}_{or}^{i^{\prime}}}^i\right)\right]\end{array}}$$

(13)

$${\displaystyle \begin{array}{l}{Obj}_3^i=\mathit{\max}\kern3.5em \left[\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_g=1\\ {}\ \end{array}}^{J_g}{Sp_g}_p^i{Nu_g}_{j_g pvt}^i+\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_{or}=1\\ {}\ \end{array}}^{J_{or}}{Sp_{or}}_p^i{Nu_{or}}_{j_{or} pvt}^i\right.\kern3.75em \\ {}\begin{array}{l}\kern10em -{Cn}^i\left(\sum_{\begin{array}{c}{r}_g=1\\ {}\ \end{array}}^{R_g}{G_{\mathrm{g}}}_{r_g}^i+\sum_{i^{\prime}\ne i}^I\sum_{\begin{array}{c}{\overline{r}}_g^{i^{\prime }}\\ {}\ \end{array}}^{{\overline{R}}_g^{i^{\prime }}}{{\overline{\overline{G}}}_g}_{{\overline{r}}_g^{i^{\prime}}}^i+\sum_{\begin{array}{c}{r}_{or}=1\\ {}\ \end{array}}^{R_{or}}{G_{or}}_{r_{or}}^i+\sum_{i^{\prime}\ne i}^I\sum_{\begin{array}{c}{\overline{r}}_{or}^{i^{\prime }}\\ {}\ \end{array}}^{{\overline{R}}_{or}^{i^{\prime }}}{{\overline{\overline{G}}}_{or}}_{{\overline{r}}_{or}^{i^{\prime}}}^i\right)\\ {}\kern10em -{Cn^{\prime}}^i\left(\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{{\overline{G}}_g}_{{\overline{r}}_g^i}^i+\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{{\overline{G}}_{or}}_{{\overline{r}}_{or}^i}^i\right)-\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{Ex_g}_{{\overline{r}}_g^{\mathrm{i}}}^i\left(1-{{\overline{G}}_g}_{{\overline{r}}_g^i}^i\right)\\ {}\begin{array}{c}\kern9.75em -\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{Ex_{or}}_{{\overline{r}}_{or}^i}^i\left(1-{{\overline{G}}_{or}}_{{\overline{r}}_{or}^i}^i\right)-\sum_{p=1}^P\sum_{\begin{array}{c}{j}_g=1\\ {}\ \end{array}}^{J_g}{Fb_g}_{j_gp}^i{Z_g}_{j_g}^i-\sum_{p=1}^P\sum_{\begin{array}{c}{j}_{or}=1\\ {}\ \end{array}}^{J_{or}}{Fb_{or}}_{j_{or}p}^i{Z_{or}}_{j_{or}}^i\\ {}\left.\kern5.75em -\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_g=1\\ {}\ \end{array}}^{J_{\mathrm{g}}}{Bu_g}_{j_gp}^i{Nu_g}_{j_g pvt}^i-\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_{or}=1\\ {}\ \end{array}}^{J_{or}}{Bu_{or}}_{j_{or}p}^i{Nu_{or}}_{j_{or} pvt}^i\right]\end{array}\end{array}\end{array}}$$

(14)

$${\displaystyle \begin{array}{l}{Obj}_3^i=\mathit{\max}\kern3.5em \left[\sum_{i=1}^I\sum_{p=1}^P\sum_{v=1}^v\sum_{\begin{array}{c}{j}_g=1\\ {}\ \end{array}}^{J_g}{Up_g}_{j_gp}{Nu_g}_{j_g pvt}^i+\sum_{i=1}^I\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_{or}=1\\ {}\ \end{array}}^{J_{or}}{Up_{or}}_{j_{or}p}{Nu_{or}}_{j_{or} pvt}^i\right.\kern3.75em \\ {}\begin{array}{l}\kern10em + Tc\left(\sum_{\begin{array}{c}j\in {\Delta }_g\cup {\Delta }_{or}\\ {}\ \end{array}}\sum_{v\in V}{d}_{0j}\left({Q}_{0 jv}+{Q}_{j0v}^{\prime}\right)+\sum_{\begin{array}{c}j,{j}^{\prime}\in {\Delta }_g\cup {\Delta }_{or}\\ {}\ \end{array}}\sum_{v\in V}{d}_{j{j}^{\prime }}{Q}_{j{j}^{\prime }v}\right)\\ {}\kern9.75em +{Pun}_{or}\left(\sum_{\begin{array}{c}j\in {\Delta }_{or}\\ {}\ \end{array}}\sum_{v\in V}{Q}_{0 jv}+\sum_{j^{\prime}\in {\Delta }_g}\sum_{j\in {\Delta }_{or}}\sum_{v\in V}{Q}_{j^{\prime } jv}+\sum_{\begin{array}{c}{j}^{\prime },j\in {\Delta }_{or},{j}^{\prime}\ne j\\ {}\ \end{array}}\sum_{v\in V}{Q}_{j^{\prime } jv}\right)\\ {}\left.\kern9.75em +\sum_{i=1}^I\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_g=1\\ {}\ \end{array}}^{J_g}{Pro}_p^g{Nu_g}_{j_g pvt}^i+\sum_{i=1}^I\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{j}_{or}=1\\ {}\ \end{array}}^{J_{or}}{Pro}_p^{or}{Nu_{or}}_{j_{or} pvt}^i\right]\end{array}\end{array}}$$

(15)

$${\displaystyle \begin{array}{l}{Obj}_4^i=\mathit{\min}\kern3.5em \left[\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{As_g^g}_{p{\overline{r}}_g^i}^i{{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^i vt}^i+\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{As_{or}^g}_{p{\overline{r}}_{or}^i}^i{{Nu^{\prime}}_{or}^g}_{p{\overline{r}}_{or}^i vt}^i\right.\kern3.75em \\ {}\kern10em +\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{As_g^{or}}_{p{\overline{r}}_g^i}^i{{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^it}^i+\sum_{\mathrm{p}=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{As_{or}^{or}}_{p{\overline{r}}_{or}^i}^i{{Nu^{\prime}}_{or}^{or}}_{p{\overline{r}}_{or}^it}^i\end{array}}$$

(16)

$${\displaystyle \begin{array}{l}{Obj_g^{ag}}_{\begin{array}{c}{\overline{r}}_g^i\\ {}\ \end{array}}^i=\mathit{\max}\kern1.25em \left[{\alpha}_g^g\sum_{p=1}^P\sum_{\begin{array}{c}{{\overline{r}}^{\prime}}_g^i\ne {\overline{r}}_g^i\\ {}\ \end{array}}^{{\overline{R}}_g^i}{Dem_g^g}_{p{{\overline{r}}^{\prime}}_g^i}^i+{\alpha}_g^{or}\sum_{p=1}^P\sum_{\begin{array}{c}{{\overline{r}}^{\prime}}_g^i\ne {\overline{r}}_g^i\\ {}\ \end{array}}^{{\overline{R}}_g^i}{Dem_g^{or}}_{p{{\overline{r}}^{\prime}}_g^i}^i\right.\kern3.75em \\ {}\kern11.75em \left.+{\alpha}_{or}^g\sum_{p=1}^P\sum_{\begin{array}{c}{\overline{r}}_{or}^i\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{Dem_{or}^g}_{p{\overline{r}}_{or}^i}^i+{\alpha}_{or}^{or}\sum_{p=1}^P\sum_{\begin{array}{c}{\overline{r}}_{or}^i\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{Dem_{or}^{or}}_{p{\overline{r}}_{or}^i}^i\right]\end{array}}$$

(17)

$${\displaystyle \begin{array}{cc}{Obj_g^1}_{\begin{array}{c}{\overline{r}}_g^i\\ {}\ \end{array}}^i=\mathit{\max}& \left[{\alpha}_g^g\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Dem_g^g}_{p{\overline{r}}_g^i}^i+{\alpha}_g^{or}\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Dem_g^{or}}_{\mathrm{p}{\overline{r}}_g^i}^i\right]\end{array}}$$

(18)

$${\displaystyle \begin{array}{l}{Obj_g^2}_{{\overline{r}}_g^i}^i=\mathit{\min}\kern1.25em \left[\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P\sum_{v=1}^V{Sp_g}_p^i{{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^i vt}^i+\sum_{p=1}^P\sum_{v=1}^V{Sp_{or}}_p^i{{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^i vt}^i\right.\kern3.75em \\ {}\kern8.25em \begin{array}{l}+\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Ac_g^g}_{p{\overline{r}}_g^i}^i{Dem_g^g}_{p{\overline{r}}_g^i}^i+\sum_{p=1}^P{Ac_g^{or}}_{p{\overline{r}}_{\mathrm{g}}^i}^i{Dem_g^{or}}_{p{\overline{r}}_g^i}^i\\ {}\left.+\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P\sum_{v=1}^V{Hc_g^g}_{p{\overline{r}}_g^i}^i\left(\frac{{{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^iv}^i}{2}\right)+\sum_{p=1}^P\sum_{v=1}^V{Hc_g^{or}}_{p{\overline{r}}_g^i}^i\left(\frac{{{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^iv}^i}{2}\right)\right]\end{array}\end{array}}$$

(19)

$${\displaystyle \begin{array}{l}{Obj}_2=\mathit{\min}\kern1.25em \left[\sum_{i=1}^I\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{R_g^i}\sum_{p=1}^P{Dn_g}_{p{\overline{r}}_g^i}\left({{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^i vt}^i+{{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^iv\mathrm{t}}^i\right)\right.\kern3.75em \\ {}\kern7.25em \begin{array}{l}+\sum_{i=1}^I\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}\sum_{p=1}^P{Dn_{or}}_{p{\overline{r}}_{or}^i}\left({{Nu^{\prime}}_{or}^g}_{p{\overline{r}}_{or}^i vt}^i+{{Nu^{\prime}}_{or}^{or}}_{p{\overline{r}}_{or}^i vt}^i\right)\\ {}\begin{array}{l}+ Tc\left(\sum_{\begin{array}{c}\overline{r}\in \\ {}\sum_i{\varphi}_g^i\cup \sum_i{\varphi}_{or}^i\\ {}\ \end{array}}\sum_{v\in V}{d}_{0\overline{r}}^{\prime}\left({U}_{0\overline{r}v}+{U}_{\overline{r}0v}^{\prime}\right)+\sum_{\begin{array}{c}\overline{r},{\overline{r}}^{\prime}\in \\ {}\sum_i{\varphi}_g^i\cup \sum_i{\varphi}_{or}^i\\ {}\ \end{array}}\sum_{v\in V}{d}_{\overline{r}{\overline{r}}^{\prime}}^{\prime }{U}_{\overline{r}{\overline{r}}^{\prime }v}\right)\\ {}\left.+{Pun}_{or}^{\prime}\left(\sum_{\begin{array}{c}\overline{r}\in \sum_i{\varphi}_{or}^i\\ {}\ \end{array}}\sum_{v\in V}{U}_{0\overline{r}v}+\sum_{\begin{array}{c}{\overline{r}}^{\prime}\in \sum_i{\varphi}_g^i\\ {}\ \end{array}}\sum_{\overline{r}\in \sum_i{\varphi}_{or}^i}\sum_{v\in V}{U}_{{\overline{r}}^{\prime}\overline{r}v}+\sum_{\begin{array}{c}{\overline{r}}^{\prime },\overline{r}\in \sum_i{\varphi}_{or}^i,\\ {}{\overline{r}}^{\prime}\ne \overline{r}\\ {}\ \end{array}}\sum_{v\in V}{U}_{{\overline{r}}^{\prime}\overline{r}v}\right)\right]\end{array}\end{array}\end{array}}$$

(20)

$${\displaystyle \begin{array}{l}{Obj_g^3}_{\begin{array}{c}{\overline{r}}_g^i\\ {}\ \end{array}}^i=\mathit{\max}\kern1.25em \left[{\lambda}_g\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Se_g}_p^i{{Nu^{\prime \prime}}_g^g}_{p{\overline{r}}_g^it}^i+{\lambda}_{or}\sum_{p=1}^P{Se_{or}}_p^i{{Nu^{\prime \prime}}_g^{or}}_{p{\overline{r}}_g^it}^i\right.\kern3.75em \\ {}\kern7.25em \begin{array}{l}\kern1.5em {\lambda}_g\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Co_g^g}_{p{\overline{r}}_g^i}^i{{Nu^{\prime \prime}}_g^g}_{p{\overline{r}}_g^it}^i+{\lambda}_{or}\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P{Co_g^{or}}_{p{\overline{r}}_g^i}^i{{Nu^{\prime \prime}}_g^{or}}_{p{\overline{r}}_g^it}^i\\ {}\begin{array}{l}\kern1.25em +{\lambda}_g\sum_{\begin{array}{c}{{\overline{r}}^{\prime}}_g^i\ne {\overline{r}}_g^i\\ {}\ \end{array}}\sum_{p=1}^P{Gr_g}_p^i{{Nu^{\prime \prime}}_g^g}_{p{{\overline{r}}^{\prime}}_g^it}^i+{\lambda}_{or}\sum_{\begin{array}{c}{{\overline{r}}^{\prime}}_g^i\ne {\overline{r}}_g^i\\ {}\ \end{array}}\sum_{p=1}^P{Gr_{\mathrm{o}r}}_p^i{{Nu^{\prime \prime}}_g^{or}}_{p{{\overline{r}}^{\prime}}_g^it}^i\\ {}\begin{array}{l}\kern1.25em +{\lambda}_g\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}\sum_{p=1}^P{Gr_g}_p^i{{Nu^{\prime \prime}}_{or}^g}_{p{\overline{r}}_{or}^it}^i+{\lambda}_{or}\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}\sum_{p=1}^P{Gr_{or}}_p^i{{Nu^{\prime \prime}}_{or}^{or}}_{p{\overline{r}}_{or}^it}^i\\ {}\begin{array}{l}\kern1.25em -\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P\sum_{v=1}^V{Pu_g^g}_{p{\overline{r}}_g^i}^i\left({{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^i vt}^i-{{Nu^{\prime \prime}}_g^g}_{p{\overline{r}}_g^it}^i\right)\\ {}\left.\kern1.25em -\sum_{\begin{array}{c}p=1\\ {}\ \end{array}}^P\sum_{v=1}^V{Pu_g^{or}}_{p{\overline{r}}_g^i}^i\left({{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^i vt}^i-{{Nu^{\prime \prime}}_g^{or}}_{p{\overline{r}}_g^it}^i\right)\right]\end{array}\end{array}\end{array}\end{array}\end{array}}$$

(21)

$${\displaystyle \begin{array}{l}{Obj}_5^i= minn\kern1.25em \left[\sum_{p=1}^P\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{Co_g^g}_{p{\overline{r}}_g^i}^i{{Nu^{\prime \prime}}_g^g}_{p{\overline{r}}_g^it}^i+\sum_{p=1}^P\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{Co_{or}^g}_{p{\overline{r}}_{or}^i}^i{{Nu^{\prime \prime}}_{or}^g}_{p{\overline{r}}_{or}^it}^i\right.\kern3.75em \\ {}\kern7.25em \begin{array}{l}\kern1em +\sum_{p=1}^P\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{Co_g^{or}}_{p{\overline{r}}_g^i}^i{{Nu^{\prime \prime}}_g^{or}}_{p{\overline{r}}_g^it}^i+\sum_{p=1}^P\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{Co_{or}^{or}}_{p{\overline{r}}_{or}^i}^i{{Nu^{\prime \prime}}_{or}^{or}}_{p{\overline{r}}_{or}^it}^i\\ {}\begin{array}{l}\kern1em +\left({\overline{R}}_g^i+{\overline{R}}_{or}^i-1\right)\left(\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}\sum_{p=1}^P{Gr_g}_p^i{{Nu^{\prime \prime}}_g^g}_{p{\overline{r}}_g^it}^i+\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}\sum_{p=1}^P{Gr_{or}}_p^i{{Nu^{\prime \prime}}_g^{or}}_{p{\overline{r}}_g^it}^i\right.\\ {}\begin{array}{l}\kern1em \left.+\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}\sum_{p=1}^P{Gr_g}_p^i{{Nu^{\prime \prime}}_{or}^g}_{p{\overline{r}}_{or}^it}^i+\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}\sum_{p=1}^P{Gr_{or}}_p^i{{Nu^{\prime \prime}}_{or}^{or}}_{p{\overline{r}}_{or}^it}^i\right)\\ {}\begin{array}{l}\kern1em -\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{Pu_{or}^g}_{p{\overline{r}}_{or}^i}^i\left({{Nu^{\prime}}_g^g}_{p{\overline{r}}_g^i vt}^i-{{Nu^{\prime \prime}}_g^g}_{p{\overline{r}}_g^it}^i\right)\\ {}\kern1em \begin{array}{l}-\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{Pu_{or}^g}_{p{\overline{r}}_{or}^i}^i\left({{Nu^{\prime}}_{or}^g}_{p{\overline{r}}_{or}^i vt}^i-{{Nu^{\prime \prime}}_{or}^g}_{p{\overline{r}}_{or}^it}^i\right)\\ {}\begin{array}{l}-\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_g^i=1\\ {}\ \end{array}}^{{\overline{R}}_g^i}{Pu_g^{or}}_{p{\overline{r}}_g^i}^i\left({{Nu^{\prime}}_g^{or}}_{p{\overline{r}}_g^i vt}^i-{{Nu^{\prime \prime}}_g^{or}}_{p{\overline{r}}_g^it}^i\right)\\ {}-\sum_{p=1}^P\sum_{v=1}^V\sum_{\begin{array}{c}{\overline{r}}_{or}^i=1\\ {}\ \end{array}}^{{\overline{R}}_{or}^i}{Pu_{or}^{or}}_{p{\overline{r}}_{or}^i}^i\left({{Nu^{\prime}}_{or}^{or}}_{p{\overline{r}}_{or}^i vt}^i-{{Nu^{\prime \prime}}_{or}^{or}}_{p{\overline{r}}_{or}^it}^i\right)\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}\end{array}}$$

(22)