A bi-objective robust optimization model for a blood collection and testing problem: an accelerated stochastic Benders decomposition

Abstract

Blood transfusion services are vital components of healthcare systems all over the world. In this paper, a generalized network optimization model is developed for a complex blood supply chain in accordance with Iranian blood transfusion organization (IBTO) structure. This structure consist of four types facilities. Blood collection centers, blood collection and processing centers, mobile teams and blood transfusion center have various duties in IBTO structure. The major contribution is to develop a novel hybrid approach based on stochastic programming, ε-constraint and robust optimization (HSERO) to simultaneously model two types of uncertainties by including stochastic scenarios for total blood donations and polyhedral uncertainty sets for demands. An accelerated stochastic Benders decomposition algorithm is proposed to solve the problem modeled in this paper. To speed up the convergence of the solution algorithm, valid inequalities are introduced to get better quality lower bounds. In addition, a Pareto-optimal cut generation scheme is used to strengthen the Benders optimality cuts. Numerical illustrations are given to verify the mathematical formulation and also to show the benefits of using the HSERO approach. At the end, the performance improvements achieved by the valid inequalities and the Pareto-optimal cuts are demonstrated in a real world application.

Appendix: The hybrid stochastic ε-constraint robust formulation

Consider the linear program (LP) in (A1), where $C$ is an $n$-vector, $A$ is a $m \times n$ matrix, and $b$ is an $m$-vector.

$$ \begin{aligned} & Min\, Cx, \\ & s.t.; \\ & Ax \le b, \\ & x \ge 0. \\ \end{aligned} $$

(A1)

Assume uncertainty only affects the elements of matrix $A$. That is, consider a particular row $I$ of $A$ and let $J_{i}$ symbolize the set of coefficients in $I$ subject to uncertainty. Each data element $\tilde{a}_{ij}$, $j \in j_{i}$ is modeled as a bounded and independent random variable taking value in an interval $\left[ {\hat{a}_{ij} - a_{ij} ,\hat{a}_{ij} + a_{ij} } \right]$, where $\hat{a}_{ij}$ is the nominal value and $a_{ij}$ is the maximum deviation from this nominal value. With this assumption, the LP in (A1) is reformulated as:

$$ \begin{aligned} & Min\, Cx, \\ & s.t.; \\ & \mathop {Max}\limits_{{\forall \tilde{a}_{ij} \in j_{i} }} \left( {\mathop \sum \limits_{j} \tilde{a}_{ij} x} \right) \le b_{i } ;\quad \forall i, \\ & x \ge 0. \\ \end{aligned} $$

(A2)

Next, we define a scaled deviation $z_{ij} = (\tilde{a}_{ij} - \hat{a}_{ij} )/a_{ij}$, in which $\tilde{a}_{ij}$, $\hat{a}_{ij}$ and $a_{ij}$ denote the uncertain value, its nominal value and its maximum deviation from the nominal value, respectively. It is unlikely that all of the uncertain input $\tilde{a}_{ij}$ and $j_{i}$ will understand their worst-case values simultaneously. Thus, a maximum number of parameters that can deviate from their nominal values for each constraint $i$ is considered as $\gamma_{i}$, called the uncertainty budget, where $\gamma_{i} \in \left[ {0,\left| {J_{i} } \right|} \right]$. The aggregated scaled deviation of uncertain parameters for constraint $i$ is bounded as $\mathop \sum \nolimits_{{j \in j_{i} }} \left| {z_{ij} } \right| \le \gamma_{i} , \forall i.$

The uncertainty budget plays a critical role in adjusting the solution’s level of conservatism against the robustness. If $\gamma_{i}$ = 0, it decreases to the nominal formulation where there is no protection against uncertainty. If $\upgamma_{{\rm i}} = \left| {J_{i} } \right|$, the ith constraint is completely kept against the worst-case realization of uncertain parameters. Finally, if $\gamma_{i} \in \left( {0,\left| {J_{i} } \right|} \right),$ then the decision maker considers a tradeoff between conservatism and cost of the solution against the level of protection against constraint violation. Based on this definition, the set $J_{i}$ is defined as $J_{i} = \left\{ {\tilde{a}_{ij} |\tilde{a}_{ij} = \hat{a}_{ij} + z_{ij} a_{ij} \;\forall i,j,z \in \varOmega } \right\}$ where $\varOmega = \left\{ {z\left| {\mathop \sum \nolimits_{j = 1}^{n} z_{ij} \le \gamma_{i} } \right.\left| {z_{ij} } \right| \le 1\;\forall i} \right\}$. Therefore;

$$ \mathop \sum \limits_{j = 1} \tilde{a}_{ij} x_{j} = \mathop \sum \limits_{j = 1} (\hat{a}_{ij} + z_{ij} a_{ij} )x_{j} = \mathop \sum \limits_{j = 1} \hat{a}_{ij} x_{j} + \mathop \sum \limits_{j = 1} z_{ij} a_{ij} x_{j} $$

(A3)

and the LP in (A2) can be reformulated as;

$$ \begin{aligned} & Min\, Cx, \\ & s.t.; \\ & \mathop \sum \limits_{j} \hat{a}_{ij} x_{j} + \mathop {Max}\limits_{{z_{i} \in \varOmega_{i} }} \left( {\mathop \sum \limits_{j} z_{ij} a_{ij} x_{j} } \right) \le b_{i } ;\quad \forall i, \\ & x \ge 0. \\ \end{aligned} $$

(A4)

The lower level problem $\mathop {Max}\nolimits_{{z_{i} \in \varOmega_{i} }} \left( {\mathop \sum \nolimits_{j} z_{ij} a_{ij} x_{j} } \right)$ for a given vector $x^{*}$ is equivalent to the following LP.

$$ \begin{aligned} & Max\,\mathop \sum \limits_{j} z_{ij} a_{ij} x_{j}^{*} , \\ & s.t.; \\ & \mathop \sum \limits_{j = 1}^{n} z_{ij} \le \gamma_{i} ;\quad \forall i, \\ & 0 \le z_{ij} \le 1;\quad \forall J \in J_{i} . \\ \end{aligned} $$

(A5)

Then, by introducing the dual variables $\alpha_{i}$ and $\beta_{ij}$, the dual of the LP in (A5) is:

$$ \begin{aligned} & Min \,\gamma_{i} \alpha_{i} + \mathop \sum \limits_{{J \in J_{i} }} \beta_{ij} , \\ & s.t.; \\ & \alpha_{i} + \beta_{ij} \ge a_{ij} x_{j}^{*} ;\quad \forall i, \\ & \alpha_{i} , J_{i} ,\beta_{ij} \ge 0;\quad \forall i. \\ \end{aligned} $$

(A6)

The dual in (A6) is applied to the LP in (A4) to obtain the robust counterpart of LP in (A1) as:

$$ \begin{aligned} & Min\,Cx, \\ & s.t.; \\ & \hat{a}_{i} x - \gamma_{i} \alpha_{i} - \mathop \sum \limits_{{J \in J_{i} }} \beta_{ij} \le b_{i } ;\quad \forall i, \\ & \alpha_{i} + \beta_{ij} \ge a_{ij} x_{j} ;\quad \forall i , \\ & \beta_{ij} , \alpha_{i} \ge 0;\quad \forall i. \\ \end{aligned} $$

(A7)

The above robust optimization model provides an efficient way to decide bounds on the probability of violation of each constraint. Let $x_{j}^{*}$ be the robust solution. Then, the violation probability of the ith constraint is calculated by:

$$ \Pr \left( {\mathop \sum \limits_{j} \bar{a}_{ij} x_{j}^{*} < b_{i} } \right) \le 1 - \emptyset \left( {\frac{{\gamma_{i} - 1}}{{\sqrt {\left| {J_{i} } \right|} }}} \right), \quad \forall i $$

(A8)

where $\emptyset \left( . \right)$ is the cumulative distribution function of the standard normal distribution. This upper bound presents a way of assigning a proper uncertainty budget to each constraint when uncertain parameters are independent and symmetrically distributed random variables in their associated uncertainty sets.

In the HRESP (Hybrid robust optimization, $\varepsilon$ constraint and stochastic programming) approach proposed in this paper for BCSND, polyhedral uncertainty sets are defined for both the blood donation volume of each category of donors and the total demand of blood in each period. To develop the uncertainty sets, the positive and the negative deviation percentages from the nominal scenarios of blood donation volume and the total demand are defined, respectively, as:

$$ \tau Bd_{i}^{ts + } = \frac{{Bd_{i}^{ts} - \widehat{Bd}_{i}^{ts} }}{{\Delta Bd_{i}^{ts + } }}\;if \;Bd_{i}^{ts} > \widehat{Bd}_{i}^{ts} ,\; \tau Bd_{i}^{ts - } = \frac{{Bd_{i}^{ts} - \widehat{Bd}_{i}^{ts} }}{{\Delta Bd_{i}^{ts - } }} \; if\;Bd_{i}^{ts} < \widehat{Bd}_{i}^{ts} $$

(A9)

$$ \tau Td_{t}^{s + } = \frac{{Td_{t}^{s} - \widehat{Td}_{t}^{s} }}{{\Delta Td_{t}^{s + } }}\;if\; Td_{t}^{s} > \widehat{Td}_{t}^{s} ,\; \tau Td_{t}^{s - } = \frac{{Td_{t}^{s} - \widehat{Td}_{t}^{s} }}{{\Delta Td_{t}^{s - } }} \;if\; Td_{t}^{s} < \widehat{Td}_{t}^{s} . $$

(A10)

Then, the uncertainty sets of blood donation volume of each category of donors and the total demand of blood in each period are:

$$ J_{s}^{Bd} = \left\{ {Bd_{i}^{ts} \left| {\begin{array}{l} {Bd_{i}^{ts} = \widehat{Bd}_{i}^{ts} +\Delta Bd_{i}^{ts + } \times \tau Bd_{i}^{ts + } -\Delta Bd_{i}^{ts - } \times \tau Bd_{i}^{ts - } } \\ {\forall k,t,\forall \tau Bd_{i}^{ts + } ,\tau Bd_{i}^{ts - } \in k^{Bd} } \\ \end{array} } \right.} \right\}, $$

(A11)

where,

$$ k^{Bd} = \left\{ {\tau Bd_{i}^{ts + } ,\tau Bd_{i}^{ts - } \left| {\begin{array}{l} {0 \le \tau Bd_{i}^{ts + } \le 1,0 \le \tau Bd_{i}^{ts - } \le 1,} \\ {\mathop \sum \limits_{k} \mathop \sum \limits_{t} \left( {Bd_{i}^{ts + } + \tau Bd_{i}^{ts - } } \right) \le \gamma_{s}^{Bd} } \\ \end{array} } \right.} \right\}. $$

(A12)

Similarly;

$$ J_{s}^{Td} = \left\{ {Td_{t}^{s} \left| {\begin{array}{l} {Td_{t}^{s} = \widehat{Td}_{t}^{s} +\Delta Td_{t}^{s + } \times \tau Td_{t}^{s + } -\Delta Bd_{i}^{ts - } \times \tau Td_{t}^{s - } \quad \forall k,t,} \\ {\forall \tau Td_{t}^{s + } ,\tau Td_{t}^{s - } \in k^{Td} } \\ \end{array} } \right.} \right\} $$

(A13)

$$ k^{Td} = \left\{ {\tau Td_{t}^{s + } ,\tau Td_{t}^{s - } \left| {\begin{array}{l} {0 \le \tau Td_{t}^{s + } \le 1,0 \le \tau Td_{t}^{s - } \le 1,} \\ {\mathop \sum \limits_{k} \mathop \sum \limits_{t} \left( {\tau Td_{t}^{s + } + \tau Td_{t}^{s - } } \right) \le \gamma_{s}^{Td} } \\ \end{array} } \right.} \right\} $$

(A14)

The parameter $\gamma_{s}^{Bd}$ in (A12) is the uncertainty budget for blood donation volume in scenario $s$ via which one can constrain the number of periods in which the blood donation volume may deviate from its nominal value. Similar definitions apply to the polyhedral uncertainty sets of the total demand in (A13). Allowing for this uncertainty implies that Constraints (32) and (35) may not be satisfied. In the proposed HRESP, these constraints are relaxed, where their violations are penalized in the objective function. The aim is to minimize the worst-case costs associated with violations of the Constraints (32) and (35). To incorporate the uncertainty sets (A12) and (A14) in the stochastic formulation (3)–(31), (33), (34), (36) the objective function terms containing random blood donation volume and total demand parameters of scenario $s$ are isolated using the following nonlinear expression:

$$ \begin{aligned} & PC_{s} \left( {BV,dist} \right) \\& \quad= \mathop {Max}\limits_{{Bd_{i}^{ts} \in J_{s}^{Bd} }} \left[ {\begin{array}{l} {\mathop \sum \limits_{t} \mathop \sum \limits_{i} \left( {Bd_{i}^{ts} - \mathop \sum \limits_{{\rm j = 1}}^{{\rm J}} {{\rm Bv}}_{{\rm ij}}^{{\rm ts}} - \mathop \sum \limits_{{\rm k = 1}}^{{\rm k}} {{\rm Bv}}_{{\rm ik}}^{{\rm ts}} - \mathop \sum \limits_{{\rm m = 1}}^{{\rm M}} {{\rm Bv}}_{{\rm im}}^{{\rm ts}} -{{\rm Bv}}_{{\rm i}}^{{\rm ts}} } \right) \times SC^{Bd} ,} \\ {\mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( {\mathop \sum \limits_{{\rm j = 1}}^{{\rm J}} {{\rm Bv}}_{{\rm ij}}^{{\rm ts}} + \mathop \sum \limits_{{\rm k = 1}}^{{\rm k}} {{\rm Bv}}_{{\rm ik}}^{{\rm ts}} + \mathop \sum \limits_{{\rm m = 1}}^{{\rm M}} {{\rm Bv}}_{{\rm im}}^{{\rm ts}} +{{\rm Bv}}_{{\rm i}}^{{\rm ts}} - Bd_{i}^{ts} } \right) \times PC^{Bd} } \\ \end{array} } \right] \\ & \quad\quad + \mathop {Max}\limits_{{Td_{t}^{s} \in J_{s}^{Td} }} \left[ {\mathop \sum \limits_{t} \left( {Td_{t}^{s} - \mathop \sum \limits_{k = 1}^{k} dist_{k}^{ts} - dist^{ts} } \right) \times SC^{Td} ,\mathop \sum \limits_{t} \left( {\mathop \sum \limits_{k = 1}^{k} dist_{k}^{ts} + dist^{ts} - Td_{t}^{s} } \right) \times PC^{Bd} } \right] \\ \end{aligned} $$

(A15)

This term represents the worst-case value for penalty, waste and surplus costs. Then, the nonlinear optimization problem is reformulated using auxiliary variables $ZZ1_{s}$ and $ZZ2_{s}$ as the following LP for each Scenario $s$:

$$ \mathop {Min}\limits_{{ZZ1_{s} ,ZZ2_{s} }} PC_{s} \left( {BV,dist} \right) = ZZ1_{s} + ZZ2_{s} $$

(A16)

$$ \begin{aligned} & s.t.; \\ & \mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( {Bd_{i}^{ts} - \mathop \sum \limits_{{\rm j = 1}}^{{\rm J}} {{\rm Bv}}_{{\rm ij}}^{{\rm ts}} - \mathop \sum \limits_{{\rm k = 1}}^{{\rm k}} {{\rm Bv}}_{{\rm ik}}^{{\rm ts}} - \mathop \sum \limits_{{\rm m = 1}}^{{\rm M}} {{\rm Bv}}_{{\rm im}}^{{\rm ts}} -{{\rm Bv}}_{{\rm i}}^{{\rm ts}} } \right) \times SC^{Bd} \le ZZ1_{s} ;\quad \forall Bd_{i}^{ts} \in J_{s}^{Bd} \\ \end{aligned} $$

(A17)

$$ \mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( {\mathop \sum \limits_{{\rm j = 1}}^{{\rm J}} {{\rm Bv}}_{{\rm ij}}^{{\rm ts}} +\mathop \sum \limits_{{\rm k = 1}}^{{\rm k}} {{\rm Bv}}_{{\rm ik}}^{{\rm ts}} + \mathop \sum \limits_{{\rm m = 1}}^{{\rm M}} {{\rm Bv}}_{{\rm im}}^{{\rm ts}} +{{\rm Bv}}_{{\rm i}}^{{\rm ts}} - Bd_{i}^{ts} } \right) \times PC^{Bd} \le ZZ1_{s} ;\quad \forall Bd_{i}^{ts} \in J_{s}^{Bd} $$

(A18)

$$ \mathop \sum \limits_{t} \left( {Td_{t}^{s} - \mathop \sum \limits_{k = 1}^{k} dist_{k}^{ts} - dist^{ts} } \right) \times SC^{Td} \le ZZ2_{s} ;\quad \forall Td_{t}^{s} \in J_{s}^{Td} $$

(A19)

$$ \mathop \sum \limits_{t} \left( {\mathop \sum \limits_{k = 1}^{k} dist_{k}^{ts} + dist^{ts} - Td_{t}^{s} } \right) \times PC^{Bd} \le ZZ2_{s} ;\quad \forall Td_{t}^{s} \in J_{s}^{Td} $$

(A20)

$$ ZZ1_{s} ,ZZ2_{s} \ge 0. $$

(A21)

The Constraints (A17)–(A21) should be satisfied for all realizations of the uncertain blood donation volume and total demand in their polyhedral uncertainty sets. We find their robust counterparts, explained in details for Constraint (A17) for instance. From the set definition (A11), the Constraint (A17) can be re-written as:

$$ \mathop {Max}\limits_{{Bd_{i}^{ts} \in J_{s}^{Bd} }} \left[ {\mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( {Bd_{i}^{ts} - \mathop \sum \limits_{{\rm j = 1}}^{{\rm J}} {{\rm Bv}}_{{\rm ij}}^{{\rm ts}} - \mathop \sum \limits_{{\rm k = 1}}^{{\rm k}} {{\rm Bv}}_{{\rm ik}}^{{\rm ts}} - \mathop \sum \limits_{{\rm m = 1}}^{{\rm M}} {{\rm Bv}}_{{\rm im}}^{{\rm ts}} -{{\rm Bv}}_{{\rm i}}^{{\rm ts}} } \right) \times SC^{Bd} \le ZZ1_{s} } \right] $$

(A22)

$$ \begin{aligned} & = \mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( {\widehat{Bd}_{i}^{ts} - \mathop \sum \limits_{{\rm j = 1}}^{{\rm J}} {{\rm Bv}}_{{\rm ij}}^{{\rm ts}} - \mathop \sum \limits_{{\rm k = 1}}^{{\rm k}} {{\rm Bv}}_{{\rm ik}}^{{\rm ts}} - \mathop \sum \limits_{{\rm m = 1}}^{{\rm M}} {{\rm Bv}}_{{\rm im}}^{{\rm ts}} -{{\rm Bv}}_{{\rm i}}^{{\rm ts}} } \right) \times SC^{Bd} \\ & \quad + \mathop {Max}\limits_{{\forall \tau Bd_{i}^{ts + } ,\tau Bd_{i}^{ts - } \in k^{Bd} }} \left[ {\mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( {\Delta Bd_{i}^{ts + } \times \tau Bd_{i}^{ts + } - \Delta Bd_{i}^{ts - } \times \tau Bd_{i}^{ts - } } \right) \times SC^{Bd} \le ZZ1_{s} } \right] \\ \end{aligned} $$

(A23)

In this constraint, we optimize over the positive and negative deviation percentages from nominal scenario for uncertain blood donation volume. The maximization problem in (A23) is expanded considering constraints from polyhedral uncertainty sets as follows:

$$ \mathop { - Min}\limits_{{\forall \tau Bd_{i}^{ts + } ,\tau Bd_{i}^{ts - } \in k^{Bd} }} \left[ {\mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( { -\Delta Bd_{i}^{ts + } \times \tau Bd_{i}^{ts + } +\Delta Bd_{i}^{ts - } \times \tau Bd_{i}^{ts - } } \right) \times SC^{Bd} } \right] $$

(A24)

$$ \begin{aligned} & s.t.; \\ & - \tau Bd_{i}^{ts + } \ge - 1\quad \forall t,i\quad :\delta 1_{i}^{ts} \\ \end{aligned} $$

(A25)

$$ - \tau Bd_{i}^{ts - } \ge - 1\quad \forall t,i \quad :\delta 2_{i}^{ts} $$

(A26)

$$ \mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( { - \tau Bd_{i}^{ts + } - \tau Bd_{i}^{ts - } } \right) \ge - \gamma_{s}^{Bd} \quad :\delta 3^{Bd} $$

(A27)

$$ \tau Bd_{i}^{ts + } ,\tau Bd_{i}^{ts - } \ge 0 $$

(A28)

Then, the dual is taken as:

$$ \mathop {Max}\limits_{{\delta 1_{i}^{ts} ,\delta 2_{i}^{ts} ,\delta 3^{Bd} }} \left[ { - \gamma_{s}^{Bd} \times \delta 3^{Bd} - \mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( {\delta 1_{i}^{ts} + \delta 2_{i}^{ts} } \right)} \right] $$

(A29)

$$ \begin{aligned} & s.t.; \\ & - \delta 1_{i}^{ts} - \delta 3^{Bd} \le -\Delta Bd_{i}^{ts + } ;\quad \forall i,t \\ \end{aligned} $$

(A30)

$$ - \delta 2_{i}^{ts} - \delta 3^{Bd} \le -\Delta Bd_{i}^{ts - } ;\quad \forall i,t $$

(A31)

$$ \delta 2_{i}^{ts} ,\delta 2_{i}^{ts} ,\delta 3^{Bd} \ge 0;\quad \forall i,t $$

(A32)

According to strong duality theory (Keyvanshokooh et al. 2016), as the Constraint (A31) is redundant, the term $\delta 2_{i}^{ts}$ can be removed. Then, (A29) is replaced without this term in Constraint (A23). Hence, the robust counterpart of Constraint (A17) is equivalent to:

$$ \begin{aligned} & \left( {\mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( {\widehat{Bd}_{i}^{ts} + \delta 1_{i}^{ts} } \right) + \delta 3^{Bd} \times \gamma_{s}^{Bd} - \mathop \sum \limits_{{\rm j = 1}}^{{\rm J}} \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{s = 1}^{S} Bv_{ij}^{ts} - \mathop \sum \limits_{{\rm k = 1}}^{{\rm k}} \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{s = 1}^{S} \mathop \sum \limits_{t = 1}^{T} Bv_{ik}^{ts} } \right. \\ & \quad \left. { - \mathop \sum \limits_{{\rm m = 1}}^{{\rm M}} \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{s = 1}^{S} \mathop \sum \limits_{t = 1}^{T} Bv_{im}^{ts} - \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{s = 1}^{S} Bv_{i}^{ts} } \right) \times SC^{Bd} \le ZZ1_{s} \\ \end{aligned} $$

(A33)

$$ \begin{aligned} & s.t.; \\ & \delta 1_{i}^{ts} + \delta 3^{Bd} \ge \Delta Bd_{i}^{ts + } ;\quad \forall i,t \\ \end{aligned} $$

(A34)

$$ \delta 2_{i}^{ts} ,\delta 3^{Bd} \ge 0;\quad \forall i,t $$

(A35)

The robust counterparts of the other constraints are found similarly.

Finally, the proposed hybrid stochastic ε-constraint robust formulation of the BSCND problem is:

$$ Min\,Zn_{1} = \mathop \sum \limits_{s} \pi_{s} (Z_{1} - ZZ1_{s} -ZZ2_{s} ) $$

(41)

s.t.;

Constraints (3)–(31), (33), (34), (36)

$$ \left( \begin{aligned} \mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( {\widehat{Bd}_{i}^{ts} + \delta 1_{i}^{ts} } \right) + \delta 3^{Bd} \times \gamma_{s}^{Bd} - \mathop \sum \limits_{{\rm j = 1}}^{{\rm J}} \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{s = 1}^{S} Bv_{ij}^{ts} - \mathop \sum \limits_{{\rm k = 1}}^{{\rm k}} \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{s = 1}^{S} \mathop \sum \limits_{t = 1}^{T} Bv_{ik}^{ts} \hfill \\ \quad - \mathop \sum \limits_{{\rm m = 1}}^{{\rm M}} \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{s = 1}^{S} \mathop \sum \limits_{t = 1}^{T} Bv_{im}^{ts} - \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{s = 1}^{S} Bv_{i}^{ts} \hfill \\ \end{aligned} \right)SC^{Bd} \le ZZ1_{s} ,\quad \forall s $$

(42)

$$ \left( \begin{aligned} \mathop \sum \limits_{i} \mathop \sum \limits_{t} \left( {\widehat{Bd}_{i}^{ts} - \delta 2_{i}^{ts} } \right) - \delta 4^{Bd} \times \gamma_{s}^{Bd} - \mathop \sum \limits_{{\rm j = 1}}^{{\rm J}} \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{s = 1}^{S} Bv_{ij}^{ts} - \mathop \sum \limits_{{\rm k = 1}}^{{\rm k}} \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{s = 1}^{S} \mathop \sum \limits_{t = 1}^{T} Bv_{ik}^{ts} \hfill \\ \quad - \mathop \sum \limits_{{\rm m = 1}}^{{\rm M}} \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{s = 1}^{S} \mathop \sum \limits_{t = 1}^{T} Bv_{im}^{ts} - \mathop \sum \limits_{i = 1}^{I} \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{s = 1}^{S} Bv_{i}^{ts} \hfill \\ \end{aligned} \right)PC^{Bd} \ge - ZZ1_{s} ,\quad \forall s $$

(43)

$$ \left( {\mathop \sum \limits_{t} \left( {\widehat{Td}_{t}^{s} + \delta 5_{s}^{t} } \right) + \delta 7^{Td} \times \gamma_{s}^{Td} - \mathop \sum \limits_{{\rm k = 1}}^{{\rm k}} \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{s = 1}^{S} dist_{k}^{ts} - \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{s = 1}^{S} dist^{ts} } \right)SC^{Td} \le ZZ2_{s} ,\quad \forall s $$

(44)

$$ \left( {\mathop \sum \limits_{t} \left( {\widehat{Td}_{t}^{s} - \delta 6_{t}^{s} } \right) - \delta 8^{Td} \times \gamma_{s}^{Td} - \mathop \sum \limits_{{\rm k = 1}}^{{\rm k}} \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{s = 1}^{S} dist_{k}^{ts} - \mathop \sum \limits_{t = 1}^{T} \mathop \sum \limits_{s = 1}^{S} dist^{ts} } \right)PC^{Td} \ge - ZZ2_{s} ,\quad \forall s $$

(45)

$$ \delta 1_{i}^{ts} + \delta 3^{Bd} \ge\Delta Bd_{i}^{ts + } ;\quad \forall i,t,s $$

(46)

$$ \delta 2_{i}^{ts} + \delta 4^{Bd} \ge\Delta Bd_{i}^{ts - } ;\quad \forall i,t,s $$

(47)

$$ \delta 5_{s}^{t} + \delta 7^{Bd} \ge\Delta Td_{t}^{s + } ;\quad \forall i,t,s $$

(48)

$$ \delta 6_{t}^{s} + \delta 8^{Bd} \ge\Delta Td_{t}^{s - } ;\quad \forall i,t,s $$

(49)

$$ ZZ1_{s} ,ZZ2_{s} ,\delta 1_{i}^{ts} ,\delta 2_{i}^{ts} ,\delta 3^{Bd} ,\delta 4^{Bd} ,\delta 5_{s}^{t} ,\delta 6_{t}^{s} ,\delta 7^{Bd} ,\delta 8^{Bd} \ge 0;\quad \forall i,t,s $$

(50)