Appendix 1: Proofs of Theorem 1 and Corollary 1
Consider an optimal solution \((\bar{x}, \bar{y}, \bar{q})\) of cost \(\bar{z}(F)\), and we define
$$\begin{aligned} J_{k}= & {} \{l\in R_{k}:\bar{x}_{kl}=1\},\quad k\in N_{S}\cup N_{D}\\ L_{k}= & {} \{r\in M_{d}:\bar{y}_{dr}=1\}, \quad d\in N_{D}\\ \bar{V}_{k}= & {} \{i\in R_{kl}:l\in J_{k}\},\quad k\in N_{S}\cup N_{D}\\ \bar{N}_{S}= & {} \{i\in R_{r}:r \in L_{k}:k\in N_{D}\\ \bar{N}_{D}= & {} \left\{ d \in N_{D}:\sum \limits _{l\in R_{d}}\bar{x}_{dl}\geqslant 1\right\} \end{aligned}$$
Let \(z(RF(\beta ,\lambda ,\mu ))\) be the optimal cost, of a valid group of \((\beta ,\lambda ,\mu )\). So, from (11), we have
$$\begin{aligned} \tilde{c}_{kl}=c_{kl}-\sum \limits _{i\in N_{c}}\alpha _{ikl}\cdot (\beta _{ik}+\lambda _{i})-\mu _{k}\geqslant 0, \quad k\in N_{S}\cup N_{D},\quad l\in R_{k} \end{aligned}$$
Then
$$\begin{aligned} \sum \limits _{k\in \bar{N}_{S}\cup \bar{N}_{D}}\sum \limits _{l\in J_{k}}\tilde{c}_{kl}=\sum \limits _{k\in \bar{N}_{S}\cup \bar{N}_{D}}\sum \limits _{l\in J_{k}}c_{kl}-\sum \limits _{k\in \bar{N}_{S}\cup \bar{N}_{D}}\sum \limits _{l\in J_{k}}\sum \limits _{i\in N_{c}}\alpha _{ikl}\cdot (\beta _{ik}+\lambda _{i})-\sum \limits _{k\in \bar{N}_{S}\cup \bar{N}_{D}}\sum \limits _{l\in J_{k}}\mu _{k} \end{aligned}$$
(30)
and obviously,
$$\begin{aligned}&\sum \limits _{l\in J_{k}}\sum \limits _{i\in N_{c}}\alpha _{ikl}\cdot (\beta _{ik}+\lambda _{i})=\sum \limits _{i\in \bar{V}_{k}}(\beta _{ik}+\lambda _{i}), k\in \bar{N}_{S}\cup \bar{N}_{D} \end{aligned}$$
(31)
$$\begin{aligned}&\sum \limits _{k\in \bar{N}_{S}\cup \bar{N}_{D}}\sum \limits _{l\in J_{k}}\mu _{k} =\sum \limits _{k\in \bar{N}_{S}\cup \bar{N}_{D}}|J_{k}|\cdot \mu _{k} \geqslant \sum \limits _{k\in \bar{N}_{S}\cup \bar{N}_{D}}m^{2}_{k}\cdot \mu _{k} \end{aligned}$$
(32)
move (31), (32) into (30), we get
$$\begin{aligned}&\sum \limits _{k\in \bar{N}_{S}\cup \bar{N}_{D}}\sum \limits _{l\in J_{k}}\tilde{c}_{kl}=\sum \limits _{k\in \bar{N}_{S}\cup \bar{N}_{D}}\sum \limits _{l\in J_{k}}c_{kl}+\sum \limits _{d\in N_{d}}\sum \limits _{r\in M_{d}}y_{dr}\cdot g_{dr}\nonumber \\&\quad -\,\left( \sum \limits _{d\in N_{d}}\sum \limits _{r\in M_{d}}y_{dr}\cdot g_{dr}+\sum \limits _{k\in \bar{N}_{S}\cup \bar{N}_{D}}\sum \limits _{i\in \bar{V}_{k}}(\beta _{ik} +\lambda _{i})+\sum \limits _{k\in \bar{N}_{S}\cup \bar{N}_{D}}m^{2}_{k}\cdot \mu _{k}\right) \end{aligned}$$
(33)
In (33), the left side \(\geqslant 0\), the first and the second terms of the right side \(=\,\bar{z}(F)\), and the remaining terms of the right side = the cost \(\tilde{z}(RF(\alpha ,\beta ,\gamma ))\) of feasible solution of
$$\begin{aligned} \tilde{y}= & {} \bar{y}\\ \xi _{ik}= & {} 1, \; i\in \bar{V}_{k},\; k\in \bar{N}_{S}\cup \bar{N}_{D};\; 0, \; otherwise\\ \tilde{q}= & {} \bar{q} \end{aligned}$$
so,
$$\begin{aligned} \sum _{k\in \bar{N}_{S}\cup \bar{N}_{D}}\sum _{l\in J_{k}}\tilde{c}_{kl}\leqslant \bar{z}(F)-z(RF(\beta ,\lambda ,\mu )) \end{aligned}$$
(34)
and
$$\begin{aligned} \bar{z}(F)-z(RF(\beta ,\lambda ,\mu ))\geqslant 0 \end{aligned}$$
It is obvious that corollary 1 hold by (34).
Appendix 2: Proof of Theorem 2
$$\begin{aligned} z(dual\;LF)= & {} max\sum \limits _{i\in N_{c}}u_{i}+\sum \limits _{k\in N_{S}\cup N_{D}}m^{2}_{k}\cdot \upsilon _{k}+\sum \limits _{k\in N_{S}\cup N_{D}}B^{2}_{k}\cdot \sigma _{k}+\sum \limits _{d\in N_{D}}m^{1}_{d}\cdot \eta _{d}\nonumber \\&+\,\sum \limits _{d\in N_{d}}\sum \limits _{d\in N_{D}}\sum \limits _{r\in M_{d}}\vartheta _{dr} \end{aligned}$$
s.t.
$$\begin{aligned}&\sum \limits _{i\in R_{sl}}\mu _{i}+\upsilon _{s}+w_{sl}\cdot \sigma _{s}-\omega _{sl}\cdot \alpha \leqslant c_{sl},\quad s\in N_{s},l\in R_{s} \end{aligned}$$
(35)
$$\begin{aligned}&\sum \limits _{i\in R_{dl}}\mu _{i}+\upsilon _{d}+\omega _{d}l\cdot \sigma _{d}\leqslant c_dl, \quad d\in N_{D},l\in R_{k} \end{aligned}$$
(36)
$$\begin{aligned}&\eta _{d}-Q_{1}\cdot \omega _{dr}+\vartheta _{dr}\leqslant g_{dr}, \quad d\in N_{d},r\in M_{d} \end{aligned}$$
(37)
$$\begin{aligned}&\alpha _{i}+\omega _{dr}\leqslant 0, \quad i\in N_{s},d\in N_{d},r\in M_{id} \end{aligned}$$
(38)
$$\begin{aligned}&\mu \in \mathbb {R}, \upsilon _{k},\sigma _{k},\alpha _{k},\eta _{k}, \omega _{kr},\vartheta _{kr}\leqslant 0 . \end{aligned}$$
(39)
1.1 Construct \((\beta ,\lambda ,\mu )\) satisfying (11)
Firstly, let
$$\begin{aligned} \mu _{k}= & {} \upsilon ^{*}_{k},\quad k\in N_{S}\cup N_{D}\\ \beta _{is}= & {} \mu ^{*}_{i}+q_{i}(\sigma ^{*}_{s}-\alpha ^{*}_{s}), \quad i\in N_{c},s\in N_{S}\\ \beta _{id}= & {} u^{*}_{i}+q_{i}\cdot \sigma _{d}^{*},\quad i\in N_{c},d\in N_{D}\\ \lambda _{i}= & {} 0,\quad i\in N_{c} \end{aligned}$$
By the definition of \(a_{ikl}\), it’s obvious that
$$\begin{aligned} \mathop {\sum }\limits _{i \in {N_c}} {a_{ikl}} \cdot {q_i} = {w_{kl}},k \in {N_S}\mathop \cup {N_D},\quad l \in {{\mathscr {R}}_k} \end{aligned}$$
then, we have
$$\begin{aligned} \mathop {\sum }\limits _{i \in {N_c}} {a_{ikl}} \cdot {\beta _{\mathrm{{ik}}}} = \mathop {\sum }\limits _{i \in {N_c}} {a_{ikl}} \cdot u_\mathrm{{i}}^* + {w_{kl}} \cdot \sigma _\mathrm{{k}}^* - {w_{kl}} \cdot \alpha _\mathrm{{k}}^*,\quad s \in {N_S},l \in {{\mathscr {R}}_k} \end{aligned}$$
and
$$\begin{aligned}&\mathop {\sum }\limits _{i \in {N_c}} {a_{idl}} \cdot {\beta _{\mathrm{{id}}}} = \mathop {\sum }\limits _{i \in {N_c}} {a_{idl}} \cdot u_\mathrm{{i}}^* + {w_{dl}} \cdot \sigma _\mathrm{{d}}^*,\;\;\;\;d \in {N_D},l \in {{\mathscr {R}}_d}\\&\mathop {\sum }\limits _{i \in {N_c}} {a_{ikl}} \cdot {\beta _{\mathrm{{ik}}}} \le {c_{kl}} - v_\mathrm{{k}}^*,\;\;\;\;k \in {N_S}\mathop \cup \nolimits ^ {N_D},l \in {{\mathscr {R}}_k} \end{aligned}$$
1.2 Prove \(max_{(\beta ,\lambda ,\mu )}\geqslant z(RF(\beta ,\lambda ,\mu ))\geqslant z(dual LF)\)
For
\(\mu \)
and
\(\upsilon \)
Let \((\xi ^{*},y^{*},q^{*})\) be the optimal solution of \(z(RF(\beta ,\lambda ,\mu ))\). By definition \((\beta ,\lambda ,\mu )\), we have
$$\begin{aligned} z\left( {RF\left( {\beta ,\lambda ,\mu } \right) } \right)= & {} \mathop {\sum }\limits _{s \in {N_s}} \mathop {\sum }\limits _{i \in {N_c}} (u_\mathrm{{i}}^* + {q_\mathrm{{i}}} \cdot \left( {\sigma _\mathrm{{s}}^* - \alpha _\mathrm{{s}}^*} \right) ) \cdot \xi _{is}^* + \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{i \in {N_c}} (u_\mathrm{{i}}^* + {q_\mathrm{{i}}} \cdot \sigma _\mathrm{{d}}^*) \cdot \xi _{id}^*\\&+\, \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} y_{dr}^* \cdot {g_{dr}} + \mathop {\sum }\limits _{i \in {N_c}} {\lambda _i} + \mathop {\sum }\limits _{k \in {N_s}\mathop \cup \nolimits ^ {N_D}} v_\mathrm{{k}}^* \cdot m_k^2\\= & {} \mathop {\sum }\limits _{i \in {N_c}} u_\mathrm{{i}}^* \cdot \left( \mathop {\sum }\limits _{s \in {N_s}} \xi _{is}^*\right) \mathrm{{ + }}\mathop {\sum }\limits _{s \in {N_s}} \sigma _\mathrm{{s}}^* \cdot \left( \mathop {\sum }\limits _{i \in {N_c}} {q_\mathrm{{i}}}\xi _{is}^*\right) \\&-\, \mathop {\sum }\limits _{s \in {N_s}} \alpha _\mathrm{{s}}^* \cdot \left( \mathop {\sum }\limits _{i \in {N_c}} {q_\mathrm{{i}}}\xi _{is}^*\right) \\&+\, \mathop {\sum }\limits _{i \in {N_c}} u_\mathrm{{i}}^* \cdot \left( \mathop {\sum }\limits _{d \in {N_D}} \xi _{id}^*\right) + \mathop {\sum }\limits _{d \in {N_D}} \sigma _\mathrm{{d}}^* \cdot \left( \mathop {\sum }\limits _{i \in {N_c}} {q_\mathrm{{i}}}\xi _{id}^*\right) \\&+\, \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} y_{dr}^* \cdot {g_{dr}} + \mathop {\sum }\limits _{k \in {N_s}\mathop \cup {N_D}} v_\mathrm{{k}}^* \cdot m_k^2 \end{aligned}$$
By (12), we have
$$\begin{aligned} \mathop {\sum }\limits _{i \in {N_c}} u_\mathrm{{i}}^* \cdot \left( \mathop {\sum }\limits _{k \in {N_s}\mathop \cup {N_D}} \xi _{ik}^*\right) = \mathop {\sum }\limits _{i \in {N_c}} u_\mathrm{{i}}^* \end{aligned}$$
For
\(\eta \)
and
\(\vartheta \)
By (37), we have
$$\begin{aligned} \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} {g_{dr}} \cdot {y^*}_{dr}\geqslant & {} \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} {\eta ^*}_d \cdot {y^*}_{dr} - \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} {Q_1} \cdot {\omega ^*}_{dr} \cdot {y^*}_{dr} \\&+ \,\mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} {\vartheta ^*}_{dr} \cdot {y^*}_{dr} \end{aligned}$$
Then, because \(\eta ^*_{d}\leqslant 0\), and by (13), we have
$$\begin{aligned} \mathop {\sum }\limits _{d \in {N_D}} \left( {\eta ^*}_d \cdot \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} {y^*}_{dr}\right) \ge \mathop {\sum }\limits _{d \in {N_D}} {\eta ^*}_d \cdot m_d^1 \end{aligned}$$
and because \(\vartheta ^*_d\leqslant 0\), we have
$$\begin{aligned} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} {\vartheta ^*}_d \cdot {y^*}_{dr} \geqslant \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} {\vartheta ^*}_d \end{aligned}$$
For
\(\sigma \)
From above, we have
$$\begin{aligned} z \left( {RF\left( {\beta ,\lambda ,\mu } \right) } \right)\geqslant & {} \mathop {\sum }\limits _{i \in {N_c}} u_\mathrm{{i}}^* + \mathop {\sum }\limits _{s \in {N_s}} \sigma _\mathrm{{s}}^* \cdot \left( \mathop {\sum }\limits _{i \in {N_c}} {q_\mathrm{{i}}}\xi _{is}^*\right) -\mathop {\sum }\limits _{s \in {N_S}} \alpha _\mathrm{{s}}^* \cdot \left( \mathop {\sum }\limits _{i \in {N_c}} {q_\mathrm{{i}}}\xi _{is}^*\right) \\&+\, \mathop {\sum }\limits _{i \in {N_c}} u_\mathrm{{i}}^* \cdot \left( \mathop {\sum }\limits _{d \in {N_D}} \xi _{id}^*\right) + \mathop {\sum }\limits _{d \in {N_D}} \sigma _\mathrm{{d}}^* \cdot \left( \mathop {\sum }\limits _{i \in {N_c}} {q_\mathrm{{i}}}\xi _{id}^*\right) + \mathop {\sum }\limits _{d \in {N_D}} {\eta ^*}_d \cdot m_d^1\\&-\, \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} {Q_1} \cdot {\omega ^*}_{dr} \cdot {y^*}_{dr} + \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} {\vartheta ^*}_{dr} \!+\! \mathop {\sum }\limits _{k \in {N_S}\mathop \cup {N_D}} v_\mathrm{{k}}^* \cdot m_k^2 \end{aligned}$$
Because \(\sigma ^*_d\leqslant 0\) and by (13), we have:
$$\begin{aligned} \mathop {\sum }\limits _{k \in {N_S}\mathop \cup {N_D}} \sigma _\mathrm{{k}}^* \cdot \left( \mathop {\sum }\limits _{i \in {N_c}} {q_\mathrm{{i}}}\xi _{ik}^*\right) \ge \mathop {\sum }\limits _{k \in {N_S}\mathop \cup {N_D}} \sigma _\mathrm{{k}}^* \cdot B_k^2 \end{aligned}$$
For
\(\alpha \)
and
\(\omega \)
By (14), we have
$$\begin{aligned} \mathop {\sum }\limits _{s \in {N_s}} \alpha _\mathrm{{s}}^* \cdot \left( \mathop {\sum }\limits _{i \in {N_c}} {q_\mathrm{{i}}}\xi _{is}^*\right)= & {} \mathop {\sum }\limits _{s \in {N_s}} \alpha _\mathrm{{s}}^* \cdot \left( {\mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_{sd}}} q_{srd}^*} \right) \\= & {} \mathop {\sum }\limits _{s \in {N_s}} \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_{sd}}} \alpha _\mathrm{{s}}^* \cdot q_{srd}^* \\= & {} \mathop {\sum }\limits _{i \in {N_s}} \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_{id}}} \alpha _\mathrm{{i}}^* \cdot q_{srd}^* \end{aligned}$$
Because \(\omega ^*_{kr}\leqslant 0\), and by (6), we have:
$$\begin{aligned} \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} ( - {\omega ^*}_{dr}{y^*}_{dr}{Q_1})\geqslant & {} \mathop {\sum }\limits _{\mathrm{{d}} \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} \left( - {\omega ^*}_{dr}\mathop {\sum }\limits _{i \in {R_{dr}}} q_{srd}^*\right) \\= & {} \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_d}} \mathop {\sum }\limits _{i \in {R_{dr}}} ( - {\omega ^*}_{dr} \cdot q_{srd}^*)\\= & {} \mathop {\sum }\limits _{d \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_{id}}} \mathop {\sum }\limits _{i \in {N_S}} ( - {\omega ^*}_{dr} \cdot q_{srd}^*) \\= & {} \mathop {\sum }\limits _{i \in {N_S}} \mathop {\sum }\limits _{k \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_{ik}}} ( - {\omega ^*}_{kr} \cdot q_{jrk}^*) \end{aligned}$$
and because (38), we have
$$\begin{aligned}&- \mathop {\sum }\limits _{k \in {N_s}} \alpha _\mathrm{{k}}^* \cdot \left( \mathop {\sum }\limits _{i \in {N_c}} {q_\mathrm{{i}}}\xi _{ik}^*\right) - \mathop {\sum }\limits _{k \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_k}} {Q_1} \cdot {\omega ^*}_{kr} \cdot {y^*}_{kr}\\&\quad \geqslant \, - \mathop {\sum }\limits _{i \in {N_s}} \mathop {\sum }\limits _{k \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_{ik}}} \alpha _\mathrm{{i}}^* \cdot q_{irk}^*\\&\qquad +\, \mathop {\sum }\limits _{i \in {N_s}} \mathop {\sum }\limits _{k \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_{ik}}} ( - {\omega ^*}_{kr} \cdot q_{jrk}^*)\\&\quad = \, - \mathop {\sum }\limits _{i \in {N_s}} \mathop {\sum }\limits _{k \in {N_D}} \mathop {\sum }\limits _{r \in {{\mathscr {M}}_{ik}}} (\alpha _\mathrm{{i}}^* + {\omega ^*}_{kr}) \cdot q_{jrk}^*\\&\quad \geqslant \,0 \end{aligned}$$
Finally, by all above, we get \(z(RF(\beta ,\lambda ,\mu ))\geqslant z(dual LF)\)
Appendix 3: Proofs of Theorem 3 and Corollary 2
We define the optimum solution of \({\phi _{krw}: z_i^*(d,r,w), \; i \in N_C}\). Let \(V\left( {d,r,w} \right) = \{ i \in N_C: z_i^*(d,r,w) > 0\}\) be the set of supplied customers, and \(\xi _{krw} = 1\) if and only if route \(r \in {{\mathscr {M}}_d}\) delivers demand w in the optimum solution.
Step 1: Prove
\(\zeta _{krw}\)
defined by
\((\bar{\xi }, \bar{y}, \bar{q})\)
is a feasible solution
Let \((\bar{\xi }, \bar{y}, \bar{q})\) be an optimum solution of \(z(RF(\beta , \lambda , \mu ) )\), and we define \(\overline{\mathscr {M}}_d = \{r \in \mathscr {M}_d: \bar{y}_{dr} = 1\}, \; \overline{V}_d = \{i \in N_C: \bar{\xi }_{dr} = 1\}\). For any \(r \in {{\mathscr {M}}_d}\), define \(\bar{w}_{dr} = \sum _{s \in R_{dr}} \bar{q}_{srd}\). For all \(\bar{\zeta }_{drw}\), set \(\bar{\zeta }_{drw} = 1\), when \(r \in {\overline{{\mathscr {M}}}_d}\) and \(w = \bar{w}_{dr}\); set \(\bar{\zeta }_{drw} = 0\), when \(r \in {\overline{{\mathscr {M}}}_d}\) but \(w \ne \bar{w}_{dr}\); set \(\bar{\zeta }_{drw} = 0\), when \(d \in {{\mathscr {M}}_d \setminus \overline{{\mathscr {M}}}_d}\). For \(\overline{N}_D = \{d \in N_D: \sum _{i \in N_C} q_i \cdot \bar{\xi }_{id} > 0 \}\), we define \(\bar{w}_d = \sum _{i \in N_C} q_i \cdot \bar{\xi }_{id}\). For all \(\bar{\zeta }_{dw}\), set \(\bar{\zeta }_{dw} = 1, \; when \; d \in {\overline{N}_D}\) and \(w \ne \bar{w}_d\); set \(\bar{\zeta }_{dw} = 0, \; when \; d \in {N_D \setminus \overline{N}_D}\).
Its’ obvious that \(\zeta _{drw}\) satisfies (18), now we prove that \(\bar{\zeta }_{drw}\) satisfies (17). By (14), we have
$$\begin{aligned} \sum _{d \in {N_D}} \sum _{i \in {N_c}} {\bar{\xi }_{id}} \cdot {q_i} + \sum \limits _{s \in {N_S}} \sum _{d \in {N_D}} \sum _{r \in {{\mathscr {M}}_{sk}}} {\bar{q}_{srd}} = \sum _{k \in {N_S} \cup {N_D}} \sum _{i \in {N_c}} {\bar{\xi }_{ik}} \cdot q_i . \end{aligned}$$
By \({\bar{q}_{srd}}, \; when \; r \in \mathscr {M}_d \setminus \overline{\mathscr {M}}_d, \, s \in N_S\), \({\bar{q}_{srd}} = 0, \; when \; r \in \overline{\mathscr {M}}_d, \; s \in N_S \setminus R_{dr}\), and the definition of \(\bar{w}_{dr}\) and \(\bar{\zeta }_{drw}\), we have
$$\begin{aligned} \sum _{s \in {N_S}} \sum _{d \in {N_D}} \sum _{r \in {{\mathscr {M}}_{sd}}} {\bar{q}_{srd}} = \sum _{\mathrm{{d}} \in {N_D}} \sum _{r \in \overline{\mathscr {M}}_d} \sum _{s \in {R_{dr}}} {\bar{q}_{srd}} = \sum _{d \in {N_D}} \mathop \sum _{r \in {{\mathscr {M}}_d}} \sum _{w \in {W_{dr}}} w \cdot {\bar{\zeta }_{drw}} \end{aligned}$$
and
$$\begin{aligned} \sum _{d \in {N_D}} \sum _{w \in {W_d}} w \cdot {\bar{\zeta }_{dw}} = \sum _{d \in {N_D}} \sum _{i \in {N_c}} {\bar{\xi }_{id}} \cdot {q_i}. \end{aligned}$$
By (12), we have
$$\begin{aligned} \sum _{s \in {N_S}} \sum _{i \in {N_c}} {\bar{\xi }_{is}} \cdot {q_i} = \sum _{i \in {N_c}} \left( {q_i} \cdot \sum _{s \in {N_S}} {\bar{\xi }_{is}}\right) \end{aligned}$$
and
$$\begin{aligned} \sum _{k \in {N_S} \cup {N_D}} \sum _{i \in {N_c}} {\bar{\xi }_{ik}} \cdot {q_i} = \sum _{i \in {N_c}} \left( {q_i} \cdot \sum _{k \in {N_S} \cup {N_D}} {\bar{\xi }_{ik}}\right) = \sum _{i \in {N_c}} {q_i} = {q_{tot}}. \end{aligned}$$
From the above 4 equations, we prove \(\bar{\zeta }_{drw}\) satisfies (17).
Step 2: Prove
\(\theta _{ikr}\)
defined by
\((\bar{\xi }, \bar{y}, \bar{q})\)
is a feasible solution
For \(i \in N_C\), we define \(\theta _{ikr}\). When \(s \in N_S, \; r \in \mathscr {M}_d \) and \(\mathscr {M}_{sd} = M_{sd}\),
$$\begin{aligned} {\theta _{isrd}} = \left\{ {\begin{array}{ll} 0, &{}\quad if \quad {\bar{\xi }_{is}} = 0\\ \frac{{{\bar{q}_{srd}}}}{{ \sum _{d \in {N_D}} \sum _{r \in {\overline{\mathscr {M}}_{sd}}} {\bar{q}_{srd}}}}, &{} \quad if \quad \bar{\xi }_{is} = 1 \end{array}} \right. \end{aligned}$$
When \(d \in N_D\),
$$\begin{aligned} {\theta _{id}} = \left\{ {\begin{array}{ll} 0, &{}\quad if \quad {\bar{\xi }_{id} = 0}\\ 1, &{}\quad if \quad \bar{\xi }_{id} = 1 \end{array}} \right. \end{aligned}$$
Then, for \(d \in N_D, r \in \overline{\mathscr {M}}_d, \; w = \bar{w}_{dr}\), we define \(\bar{z}_i (r, \bar{w}_{dr}) = \sum _{s \in R_{dr}} \theta _{isrd}\) , and for \(d \in N_D, \; w = \bar{w}_d\), we define \(\bar{z}_i (\bar{w}_{d}) = \theta _{id}\). It’s obvious that \(0 \leqslant z_i \leqslant 1\), now we prove \(\bar{z}_i\) satisfies the constraint \(\sum _{i \in N_C} q_i \cdot z_i = w\).
Part 1
For \(d \in N_D, r \in \overline{\mathscr {M}}_d, \; w = \bar{w}_{dr}\), we have
$$\begin{aligned} \sum _{i \in {N_C}} {q_i} \cdot {\bar{z}_i}\left( {r,{\bar{w}_{dr}}} \right) = \sum _{i \in {N_C}} {q_i} \cdot \sum _{s \in {R_{dr}}} {\theta _{isrd}}. \end{aligned}$$
By the definition of \(\theta _{isrd}\), it is easy to have
$$\begin{aligned} \sum _{i \in {N_C}} {q_i} \cdot \sum _{s \in {R_{dr}}} {\theta _{isrd}}= & {} \sum _{i \in {N_C}} \sum _{s \in {R_{dr}}} {q_i} \cdot {\theta _{isrd}} = \sum _{s \in {R_{dr}}} \sum _{i \in {\overline{V}_s}} {q_i} \cdot {\theta _{isrd}} \\= & {} \sum _{s \in {R_{dr}}} \left( \sum _{i \in {\overline{V}_s}} {q_i}\right) \cdot \frac{{{\bar{q}_{srd}}}}{{\sum _{d \in {N_D}} \sum _{r \in {\overline{\mathscr {M}}_{sd}}} {\bar{q}_{srd}}}}. \end{aligned}$$
Since \(\sum _{d \in {N_D}} \sum _{r \in {\overline{\mathscr {M}}_{sd}}} {\bar{q}_{srd}} = \sum _{i \in {\overline{V}_s}} {q_i}\), we have
$$\begin{aligned} \sum _{s \in {R_{dr}}} \left( \sum _{i \in {\overline{V}_s}} {q_i}\right) \cdot \frac{{{\bar{q}_{srd}}}}{{\sum _{d \in {N_D}} \sum _{r \in {\overline{\mathscr {M}}_{sd}}} {\bar{q}_{srd}}}} = \sum _{s \in {R_{dr}}} {\bar{q}_{srd}} = {\bar{w}_{dr}}. \end{aligned}$$
From the above 3 equations, we get
$$\begin{aligned} \sum _{i \in {N_C}} {q_i} \cdot {\bar{z}_i}\left( {r,{\bar{w}_{dr}}} \right) = {\bar{w}_{dr}}. \end{aligned}$$
Part 2
For \(d \in N_D, \; w = \bar{w}_d\), by the definition of \(\theta _{id}\), it is easy to have
$$\begin{aligned} \sum _{i \in {N_C}} {q_i} \cdot {\bar{z}_i}\left( {r,{\bar{w}_{d}}} \right) = \sum _{i \in {N_C}} {q_i} \cdot {\theta _{id}} = \sum _{i \in {\overline{V}_C}} {q_i} \cdot {\theta _{id}} = {\bar{w}_{d}}. \end{aligned}$$
Step 3: Prove
\(z(RF({\beta }, {\lambda }, {\mu })) \geqslant z(\overline{RF}(\bar{\beta }, \bar{\lambda }, \bar{\mu }))\)
By the definition of \(\theta _{isrd}\) and \(\theta _{id}\) , it is easy to know that, when i is served by \(s \in N_S, \bar{\xi }_{is} = \sum _{d \in N_D} \sum _{r \in \overline{\mathscr {M}}_{sk}} \theta _{isrd}\); When i is served by \(d \in N_D, \; \bar{\xi }_{id} = \theta _{id}\). So, we have
$$\begin{aligned} \bar{z}(RF({\beta }, {\lambda }, {\mu }))= & {} \min \sum _{k \in {N_S} \cup {N_D}} \sum _{i \in {N_c}} {\bar{\xi }_{ik}} \cdot {\beta _{ik}} + \sum _{d \in {N_D}} \sum _{r \in {{\mathscr {M}}_d}} {\bar{y}_{dr}} \cdot {g_{dr}} \\&+\, \sum _{i \in {N_c}} {\lambda _i} + \sum _{k \in {N_s} \cup {N_D}} {\mu _k} \cdot m_k^2\\= & {} \min \sum _{s \in {N_S}} \sum _{i \in {N_c}} {\bar{\xi }_{ik}} \cdot {\beta _{is}} + \sum _{d \in {N_D}} \sum _{i \in {N_c}} {\bar{\xi }_{id}} \cdot {\beta _{id}} + \sum _{d \in {N_D}} \sum _{r \in {{\mathscr {M}}_d}} {\bar{y}_{dr}} \cdot {g_{dr}} \\&+\, \sum _{i \in {N_c}} {\lambda _i} + \sum _{k \in {N_s} \cup {N_D}} {\mu _k} \cdot m_k^2\\= & {} \min \sum _{s \in {N_S}} \sum _{i \in {N_c}} \left( {{\sum _{d \in {N_D}} \sum _{r \in {\overline{\mathscr {M}}_{sd}}} {\theta _{srd}}}}\right) \cdot {\beta _{is}} + \sum _{d \in {N_D}} \sum _{r \in {\overline{{\mathscr {M}}}_d}} {g_{dr}} \\&+\, \sum _{d \in {N_D}} \sum _{i \in {N_C}} {\bar{\theta }_{id}} \cdot {\beta _{id}} + \sum _{i \in {N_c}} {\lambda _i} + \sum _{k \in {N_s} \cup {N_D}} {\mu _k} \cdot m_k^2 \end{aligned}$$
It’s obvious that:
$$\begin{aligned} \sum _{s \in {N_s}} \sum _{i \in {N_c}} \left( {\sum \limits _{d \in {N_D}} \sum _{r \in {\overline{\mathscr {M}}_{sd}}} {\theta _{isrd}}} \right) \cdot {\beta _{is}}= & {} \sum _{s \in {N_s}} \sum _{i \in {N_c}} \sum _{d \in {N_D}} \sum _{r \in {\overline{\mathscr {M}}_{sd}}} {\theta _{isrd}} \cdot {\beta _{is}}\\= & {} \sum _{d \in {N_D}} \sum _{i \in {N_c}} \sum _{s \in {N_s}} \sum _{r \in {\overline{\mathscr {M}}_{sd}}} {\theta _{isrd}} \cdot {\beta _{is}}.\\= & {} \mathop \sum _{d \in {N_D}} \sum _{i \in {N_c}} \sum _{r \in {\overline{\mathscr {M}}_d}} \sum _{s \in {R_{dr}}} {\theta _{isrd}} \cdot {\beta _{is}} \\= & {} \sum _{d \in {N_D}} \sum _{r \in {\overline{\mathscr {M}}_d}} \left( \sum _{i \in {N_c}} \sum _{s \in {R_{dr}}} {\theta _{isrd}} \cdot {\beta _{is}}\right) . \end{aligned}$$
By \({\bar{z}_i} (r, \bar{w}_{dr}) = \sum _{s \in {R_{dr}}} {\theta _{isrd}}, \; \bar{z}_i(\bar{w}_d ) = \theta _{id} = \bar{\xi }_{id}\), and the definition of \(\zeta _{drw}\) and \(\zeta _{dw}\), we have
$$\begin{aligned} \sum _{i \in {N_c}} \sum _{s \in {R_{dr}}} {\beta _{is}} \cdot {\theta _{isrd}}\ge & {} \sum _{i \in {N_c}} \left( { {\min }_{s \in {R_{dr}}} {\beta _{is}}} \right) \cdot \left( { \sum _{s \in {R_{dr}}} {\theta _{isrd}}} \right) \\= & {} \sum _{i \in {N_c}} \left( { {\min }_{s \in {R_{dr}}} {\beta _{is}}} \right) \cdot {\bar{z}_i}(r, \bar{w}_{dr}) = {\bar{\phi }_{dr{\bar{w}_{dr}}}} \end{aligned}$$
and
$$\begin{aligned} \sum _{i \in {N_c}} \sum _{s \in {R_{dr}}} {\beta _{id}} \cdot {\theta _{id}} = \sum _{i \in {N_C}} {\bar{z}_i}(\bar{w}_{d}) \cdot \beta _{id} = {\bar{\phi }_{d{\bar{w}_{d}}}}, \end{aligned}$$
so,
$$\begin{aligned} \bar{z}(RF({\beta }, {\lambda }, {\mu })) \geqslant \min \sum _{d \in {N_D}} \sum _{r \in {{\mathscr {M}}_d}} ({\bar{\phi }_{dr{\bar{w}_{dr}}}} + g_{dr})+ {\bar{\phi }_{d{\bar{w}_{d}}}} + \sum _{i \in {N_c}} {\lambda _i} + \sum _{k \in {N_s} \cup {N_D}} {\mu _k} \cdot m_k^2. \end{aligned}$$
Finally, we conclude that
$$\begin{aligned} {z}(RF({\beta }, {\lambda }, {\mu })) = \bar{z}(RF({\beta }, {\lambda }, {\mu })) \geqslant \bar{z}(\overline{RF}({\beta }, {\lambda }, {\mu })) \geqslant {z}(\overline{RF}({\beta }, {\lambda }, {\mu })). \end{aligned}$$
