key: cord-0763551-55fizlqu authors: Li, Min; Zhang, Chao; Ding, Mingxv; Lv, Ruipu title: A Two-stage Stochastic Variational Inequality Model for Storage and Dynamic Distribution of Medical Supplies in Epidemic Management date: 2021-10-01 journal: Appl Math Model DOI: 10.1016/j.apm.2021.09.033 sha: 2945ca38e75c1a654e618c9f0f3f03e405793939 doc_id: 763551 cord_uid: 55fizlqu The storage and distribution of medical supplies are important parts of epidemic prevention and control. This paper first proposes a new nonsmooth two-stage stochastic equilibrium model of medical supplies in epidemic management. The first stage addresses the storage in the pre-disaster phase, and the second stage focuses on the dynamic distribution by enrolling competitions among multiple hospitals over a period of time in the post-disaster phase. The uncertainties are the numbers of infected people treated in multiple hospitals during the period of time, which are time-varying around a nominal distribution predicted by historical experience. The two-stage stochastic equilibrium model is further approximated and transformed to a monotone two-stage stochastic variational inequality (SVI) model that is computationally tractable, with the aid of a smooth approximation technique. We employ the progressive hedging method (PHM) to solve a case study in the city of Wuhan in China suffered from the COVID-19 pandemic. Numerical results are presented to demonstrate the effectiveness of the proposed model in planning the storage and dynamic distribution of medical supplies in epidemic management. In this paper, we focus on the storage of medical supplies before an epidemic occurs, and the dynamic distribution of medical supplies after an epidemic really outbreaks. The government is the decision maker for the storage plan before an epidemic occurs, by taking consideration of the competitions of the hospitals under all possible scenarios of an epidemic. The government is considered to be the leader to determine the storage plan, and the hospitals are the followers to compete each other for providing possible dynamic distribution plans after the storage plan has been given and the scenario of the epidemic has been grasped. The government eventually decides an optimal dynamic distribution plan for the whole society among all the possible dynamic distribution plans that can be obtained from the competitions of the hospitals. In an epidemic, the number of infected people that are treated in every hospital every day is difficult to predict accurately such that decisions have to be taken under uncertainty. Moreover, it is better to consider the dynamic distribution over a period of time, rather than the static distribution only once. Because the effective treatment of infected people, the prevention and control of an epidemic, and the increasing of production capacity for medical supplies all need a key period which lasts two weeks or even longer, and the demand of medical supplies of a day depends also on distributions of the previous days. Few studies have explored the dynamic decision problems over a period [1, 2, 3, 4, 5, 6] . When an epidemic occurs, without storing medical supplies pre-disaster, insufficient or delayed medical supplies cannot meet the demand and will lead to heavy loss of life. On the other hand, storing too many medical supplies predisaster will lead to huge economic losses. Therefore, how to store pre-disaster and distribute medical supplies post-disaster of an epidemic appropriately is a challenging task. Under such circumstances, two-stage stochastic programming (SP) is one of the most popular modeling approaches to support the decision making process, see [7, 8, 9, 10, 11] , as it allows the modeler to represent the pre-and post-disaster phase together via the first-and second-stage decision variables. Another important issue is that an unfair distribution of supplies will make humans' lives different and will also cause social confliction. It is a natural requirement that the distribution of medical supplies should be fair, which means that there should be no privilege for certain groups of individuals. Fair distribution is usually captured by the concepts of equity [12] , including equitability and balance. Equitability means servicing a set of indistinguishable entities. Balance means servicing a set of entities while they are different from each other regarding their demands, claims, and preferences. In balance, an ideal solution may give each entity a different proportion of the total assignment. The proportion, however, is given by disaster managers, rather than decided by agents. In this paper, we concentrate on a certain period of time that the number of infected people is high, and the hospitals are extremely short of medical supplies. The nominal values of the numbers of infected people during the period are assumed to be known ahead based on historical information. There are finite possibilities of the unknown numbers of infected people that are within a band around the nominal values. We first develop a two-stage stochastic equilibrium model. The first stage determines the storage plan that has to be made before an epidemic occurs. It tries to minimize the storage costs as well as the costs occurred from all possible scenarios of distributions to hospitals in the second stage. The second stage is a non-cooperative multi-agent game model that determines the dynamic distribution of medical supplies based on time-variant demand. In the second stage, each hospital competes to minimize its own costs composed of transportation costs, purchase costs, as well as penalty costs if the demand is unmet during the period of time. The term of penalty costs indeed plays an important role in fairness. There are two main difficulties of solving the two-stage stochastic equilibrium model. Firstly, the minimization problem of one hospital contains the other hospitals' strategies, not yet known at the decision horizon. Thus the minimization problem cannot be solved directly. One hospital's objective is in conflict with the others'. Thus one hospital has to make its decision by taking other hospitals' decisions into account. Hospitals compete to reach a steady state. That is, no hospital can further cut down its costs by unilaterally changing its own strategy for distribution. Secondly, the term of penalty costs in the objective function of the minimization problem for each hospital involves the max operator that is nonsmooth. In order to overcome the above two difficulties, we use a smoothing function to approximate the term of penalty costs. We then transform the optimistic version of the two-stage stochastic equilibrium model into an equivalent two-stage SP, and then transform the two-stage SP equivalently into a two-stage SVI. It is worth mentioning that the multi-stage SVI was introduced by Rockafellar and Wets [13] in 2017, which is powerful to model general multi-stage SP problems and equilibrium problems. The two-stage SVI model in this paper is shown to be monotone and is solved by the PHM [14] proposed by Rockafellar and Sun in 2019, and the efficient semismooth Newton method is adopted for solving the problem corresponding to each scenario in the PHM. Numerical results demonstrate the effectiveness of the two-stage SVI model in describing the storage and dynamic distribution of medical supplies in epidemic prevention and control. The main contributions of the paper are summarized as follows. • We develop a two-stage stochastic equilibrium model for the storage and dynamic distribution of medical supplies in a period of time. • We use a smoothing function to approximate the term of penalty costs in the nonsmooth two-stage stochastic equilibrium model, and transform its optimistic version into an equivalent smooth monotone two-stage SVI model that is computationally tractable. The PHM is employed to solve the two-stage SVI model, which has global convergence property. The subproblem for each scenario is solved by the semismooth Newton method. • A realistic application to the city of Wuhan in China that was suffered from COVID-19 is considered in numerical experiments, which demonstrates that our proposed two-stage SVI model solved by the PHM provides good decisions of the storage and dynamic distribution of medical supplies. The rest of the paper is organized as follows. In Section 2, the relevant literature is reviewed. Section 3 develops the proposed two-stage stochastic equilibrium model. In Section 4, we present its smooth approximation and transform its optimistic version equivalently to a smooth two-stage SP. We then transform the smooth two-stage SP equivalently to a monotone two-stage SVI. Section 5 presents the PHM with the updating rule for the smoothing parameter to solve the two-stage SVI model. Section 6 implements the model on real data in Wuhan suffered from the COVID-19 pandemic and analyzes the important role of the penalty in fairness. The conclusions and directions for future research are provided in the final section. the disaster occurs, and then reacting once the uncertainty has been revealed. Barbarosoǧlu and Arda [8] introduced for the first time a two-stage SP model for both the pre-disaster and post-disaster stage where the supply capacities and demands are considered as random variables captured by a set of scenarios. Liu et al. [15] modeled the network retrofit problem as a two-stage SP to optimize a mean-risk objective of the system losses. Fan and Liu [16] formulated a twostage SP with equilibrium constraints on pre-disaster transportation network protection problems against uncertain future disasters. They demonstrated the applicability of the progressive hedging-based method for solving the two-stage SP model. Mete and Zabinsky [17] developed a two-stage SP for the storage and distribution of medical supplies in disaster management. Noyan [10] considered a risk-averse two-stage SP model for disaster management and discussed the importance of incorporating a risk measure to derive optimal decisions. Two decomposition algorithms based on the generic Benders decomposition approach are constructed to solve such problems. We refer to the excellent survey papers [9] , [11] and the references therein for more literature of two-stage SP models for disaster management. Dynamic distribution of emergency supplies is important. It is necessary to adjust the plans incorporating new information along time horizons. Rawls and Turnquist [1] considered the dynamic allocation to satisfy short-term demands for emergency supplies. Bozorgi-Amiri and Khorsi [6] considered a multiobjective dynamic SP model by integrating pre-disaster plans and post-disaster decisions. The above papers mainly dedicated to the short-term dynamic distribution of emergency supplies. Little attention has been paid to the long-term preparedness of emergency supplies. Yang et al. [2] proposed a distributionally robust optimization model for the dynamic distribution of emergency supplies under demand uncertainty within a relatively long period of time. Fairness is important for medical supplies. The balance concept related to fairness is often considered in various real life allocation problems [18] . In a disaster, different locations have different priorities because of different amounts of population and degrees of severity suffered from the disaster [19] , [20] . Hence, balance should be satisfied among demand points in the distribution of relief commodities for preventing a possible social disaster [21] . Bertsimas et al. [22] discussed different fairness concepts that are used to ensure fair allocation of resources. They also focused on balancing efficiency and equity in resource allocation settings [23] . Bertsimas et al. [24] proposed a modeling framework for general dynamic resource allocation problems where there is a concern of equitably distributing the delay among the resource requests. Cayirli and Veral [25] stated that fairness across patients need to be considered while designing appointment systems. Turkcan et al. [26] introduced a constrained model for sequential clinical scheduling. The proposed unfairness measures are based on the expected waiting times at each slot and the number of patients in the system at the beginning of each slot. Two-stage SVI has wide applications in economics, traffic network, electricity markets, supply chain problems, finance, and risk management under uncertain environment. Rockafellar and Wets [13] first introduced the formal definition of multi-stage SVIs in 2017. Two-stage SVI involves making a "here-and-now" decision at present to meet the uncertainty that is revealed later. This is one of the motivations of both two-stage SP and two-stage SVI. Chen, Pong and Wets [27] investigated a two-stage SVI model, and an expected residual minimization procedure is used to formulate the two-stage SVI into a two-stage SP with recourse. Chen et al. [28] provided a discrete scheme of two-stage stochastic linear complementarity problem, a special case of two-stage SVI, where the underlying random data are continuously distributed. Chen et al. [29] investigated the sample average approximation of two-stage stochastic generalized equation, which includes two-stage SVI as a special case. Rockafellar and Wets [30] developed the PHM for solving multi-stage convex SP for the first time in 1991. In 2019, Rockafellar and Sun [14] proposed the PHM for solving monotone multistage SVIs with a finite collection of scenarios for the random vector and showed the convergence of the PHM. In 2020, Rockefallar and Sun [31] considered the multistage Lagrangian SVI problem. In 2020, Li and Zhang [32] considered a two-stage SVI arising from a general convex two-stage SP, where the random variables have continuous distributions. We outline below some references that relate to the two-stage SVIs arising from non-cooperative multi-agent games. In 2017, Pang et al. [33] formally introduced and studied a non-cooperative multi-agent game under uncertainty and focused mainly on a two-stage setting of the game where each agent is risk-averse. Jiang et al. [34] in 2019 developed a two-stage SVI model for a production and supply competition of a homogenous product under uncertainty in an oligopolistic market. Zhang et al. [35] in 2019 studied a model of two-stage N-player non-cooperative game under uncertainty and showed that it is equivalent to a SVI model. The PHM is employed to solve the monotone SVI, as well as the nonmonotone case where an elicitability condition holds with convergence guarantee. We refer to the excellent survey paper by Sun and Chen [36] in 2021 for more literature of the theory, algorithms and applications for two-stage SVIs. We first propose a new two-stage stochastic equilibrium model for the storage and dynamic distribution of medical supplies in an epidemic within a period of time. In the first stage, the government needs to decide the amounts of storage of the different types of medical supplies in the hospitals before an epidemic occurs. The manager tries to minimize the total cost of hospitals, including the sum of the storage costs, and the costs of all hospitals in the second stage for the dynamic distribution during the period of time, subject to several constraints such as the budget limits for epidemic prevention, and the capacities of storage of the hospitals, etc. Storing a large amount of medical supplies ahead is beneficial for timely and effective aid to a sudden epidemic. However, this strategy may become a burden of social economy. It is challenging to find a good trade-off between maximizing the demand coverage and minimizing the total cost in the first stage. The demands of a hospital for all kinds of medical supplies on one day are considered as stochastic variables, which are not known before this day. The decisions of the first stage have to be made without grasping the realizations of the stochastic variables. However, it is reasonable to assume that the nominal values of the stochastic variables during the period of time we considered are known with the aid of historical experience. In the second stage, the realizations of the stochastic demand vector are grasped. The government has to decide further the distribution of medical supplies for each hospital according to the storage capacities, the actual demands, as well as the other hospitals' decisions. The medical supplies are classified into two categories -government distribution and social non-targeted donation. The social non-targeted donation refers to the medical supplies donated from organizations, companies and individuals for the epidemic, without specifying the medical supplies to which hospitals. The hospitals only need to pay the transportation costs for these medical supplies. The goal of the second stage of a hospital is to minimize the total cost of its own. The total cost of a hospital include the transportation costs, the purchase costs, and the penalty costs for unmet demands. The overall effects of the second-stage decisions under all possible scenarios are enrolled in the costs of all hospitals in the first-stage objective function. To build the two-stage stochastic equilibrium model for the storage and dynamic distribution of medical supplies in this section and also transform the model in the next section, we introduce some notations as follows. Note: Here x ij max , y jt max , z jt max > 0 for all i ∈ I, j ∈ J , t ∈ T . There may exist x j max = 0 for some j, since some medical supply is only available after the epidemic occurs, e.g., the inspection and testing articles for the COVID-19 pandemic. Let us also define the decision vectors for the two stages as follows. Because the decision vector of hospital i is closely related to the decision vectors of the other hospitals, for i = 1, 2, . . . , m, we define the vectors and we add superscript * to the above vectors for the decision vectors of the other hospitals in an optimal solution. For instance, is the decision vector for hospitals except hospital i in an optimal solution. The optimal solution of our two-stage stochastic equilibrium problem will be given in detail below. From the above discussion, we can formulate a two-stage stochastic equilibrium problem as follows. The total cost includes the acquisition and holding costs, as well as the transportation costs, the government purchasing costs and the penalty costs for unmet demands. These costs reside in two phases: the storage phase and the distribution phase. In the storage phase, we denote by m i=1 n j=1 a ij x ij the acquisition and holding costs, and denote by horizons in the distribution phase. Noting that medical supplies are classified into non-reusable and reusable medical supplies, we use M ijt (ξ) to represent the amount of unmet demands of hospital i for medical supply j on day t, which is defined as Here, we assume that reusable medical supplies do not expire within the time horizons considered. A penalty is added into the model to reduce a huge loss of life and wealth after an epidemic. The term of penalty costs is defined as n j=1 l t=1 c ij (M ijt (ξ)) + for hospital i, where the notation (ρ) + := max(ρ, 0) for any ρ ∈ R. Based on the above description, we obtain the following objective function for the first stage where for each i ∈ I, ϑ i (x i , x −i * , ξ) is the optimal value of the second-stage equilibrium problem. The objective function of the second-stage equilibrium problem is given in (3) and the constraints are described in (6)- (8) and (10) below. The objective function (2) x ij ≤ x ij max , ∀i ∈ I, j ∈ J . Constraints (6) and (7) limit the amounts of medical supplies distributed by government and obtained from social non-targeted donation for almost every (a.e.) realization of ξ ∈ Ξ, respectively. That is, for a.e. ξ ∈ Ξ, Constraint (8) ensures that the amount of medical supply j from storage used by hospital i over all time horizons does not exceed the total amount of storage. That is, for any i ∈ I, and for a.e. ξ ∈ Ξ, Constraints (9) and (10) are nonnegativity conditions, since the amount of storage and the amount of distribution cannot be negative. That is, 13 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 and for any i ∈ I, and for a.e. ξ ∈ Ξ, We construct the two-stage stochastic equilibrium model in the following where ϑ i x i , x −i * , ξ is the optimal value of the second-stage equilibrium problem. That is, for every hospital i ∈ I, Here the objective function The model (11)-(12) is a two-stage stochastic equilibrium model, where the second-stage problem is nonsmooth since the penalty in the objective function involves the max operator. The methods for solving the two-stage SP are not applicable for the two-stage stochastic equilibrium model. Hence it is computationally intractable. In the next section, we will replace the nonsmooth penalty by its smooth counterpart and transform the smooth approximation of the twostage stochastic equilibrium problem to a monotone two-stage SVI problem that is computationally tractable. For simplicity, we always assume the support set Ξ being a finite set of "scenarios" ξ k , k = 1, 2, . . . , K, each having a nonzero probability τ k such that K k=1 τ k = 1 at follows. In fact, if ξ ∈ Ξ is continuously distributed, we can adopt the sample average approximation to obtain a discrete distribution that approximates the continuous distribution as in [28, 29] . 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 In order to construct the smooth approximation of F i (y i (ξ), z i (ξ), s i (ξ)) for any ξ ∈ Ξ, we need the following definition for smoothing function. Definition 1. [37] Let g : R n → R be a locally Lipschitz continuous function. We callg : R n × R ++ → R a smoothing function of g ifg is continuously differentiable on R n for any µ ∈ R ++ and for any x ∈ R n , We use the smoothing functionq : R × R ++ → R as in [38, 39] q(ρ, µ) = µ ln e ρ/µ + 1 , to approximate the nonsmooth function ρ + for ρ ∈ R. It is easy to check that and for each fixed smoothing parameter µ > 0, the first-order derivative and the second-order derivative ofq with respect to ρ arẽ Thus given a fixed smoothing parameter µ > 0, we obtain the smooth approxi- the second stage of the two-stage stochastic equilibrium model for any hospital For each fixed µ > 0, since the second-order derivative ofq(·, µ) in (15) is greater than zero, it follows thatq(·, µ) is a convex function. Moreover, for each According to Theorem 3.1.6 of [40] , the composition of a convex function and an affine function is again a convex function. 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 This, together with the fact that the first two terms inF i (y are linear functions, and c ij > 0 for all i, j, The smooth approximation of the two-stage stochastic equilibrium model whereθ i x i , x −i * , ξ k , µ is the optimal value of the smooth approximation of the second-stage equilibrium problem. That is, for every hospital i ∈ I, it solves the following smooth minimization problem The second-stage problem (18) for hospital i involves other hospitals' optimal strategies, and hence it is not solvable itself. We thus consider the following model defined in (19) . More details on the equivalency of (17)- (18) and (19) have been provided in Subsection 4.1. We have already shown before thatF i (y function. The above model is a convex model with linear constraints. Moreover, 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 the feasible region defined by the linear constraints is compact. Since a continuous function is sure to obtain its minimum in a nonempty and compact set, the existence of solutions of the convex model (19) is guaranteed. The linear constraints also guarantee that any solution of the convex model (19) is a KKT point. In order to express the formulas of the KKT systems involving in (17)- (18) as well as (19) in a concise way, we use the vectors and matrices for the objective functions and the constraints, as well as the corresponding Lagrange multipliers for the constraints. For given positive integers α and β, we denote by I α the α×α identity matrix, by I Now assume that the first n 1 medical supply types are non-reusable and the remaining n 2 medical supply types are reusable, i.e., n 1 + n 2 = n. Let H 1 ∈ R l×l be an upper triangular matrix whose upper diagonal entries are all 1, 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 The convex model (19) can be rewritten in the form We denote by λ ∈ R mn+n + , π y (ξ k ) ∈ R nl + , π z (ξ k ) ∈ R nl + , and π s (ξ k ) ∈ R mn + the vectors of Lagrange multipliers of the first four groups of functional constraints in (21) , respectively. Using the above notations, the KKT system of (21) can be written as the following two-stage SVI model Here 0 ≤ u⊥v ≥ 0 means that u ≥ 0, v ≥ 0, and u T v = 0, for two vectors u and v of the same dimension. The vectors x and λ are the first-stage decision vectors of the above two-stage SVI model, which should be given before the scenario ξ k really happens. That is, x and λ satisfy the "nonanticipativity" property as mentioned in [13] . The vectors are the second-stage decision vectors which should be given after we know the realization of ξ k . It is worth mentioning that (22) is a two-stage stochastic 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 nonlinear complementarity problem (NCP), which is a special type of two-stage SVIs. The nonlinearity comes from M ξ k µ ijt in defining Q ξ k µ yzs . Proposition 1. The two-stage SVI (22) is of maximal monotone type. Proof. See Appendix A.1 for the proof. In this subsection, we discuss the relation of the two-stage stochastic equilibrium model (17)- (18) and the two-stage SP model (19) . The two-stage stochastic equilibrium model (17)-(18) can be written as whereθ i x i , x −i * , ξ k , µ is the optimal value of the second-stage equilibrium problem. That is, for every hospital i ∈ I, it solves the following smooth minimization problem Note that constraint (8) 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 if there exist vectorŝ together with the vector y(ξ k ) T , z(ξ k ) T , s(ξ k ) T , k = 1, . . . , K T , satisfy the following NCP. Proposition 2. For any feasible solution x ∈ R mn of (17), the solution set (18) is nonempty, convex, and compact. Proof. See Appendix A.2 for the proof. There may exist many choices of decisions in S 2 (x) for a given x ∈ R mn . It is reasonable to assume that the government as the leader for the two-stage equilibrium model is allowed to select the decision vector such that the total cost involved in the second stage is minimized. That is, 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Similar as bilevel programming [41] , by assuming (27), we indeed consider the optimistic version of the two-stage stochastic equilibrium model (17) The solution set of (28) is nonempty, since the feasible set of (28) is nonempty, convex, and compact, and the objective function is continuous. Proposition 3. The two-stage SP model (19) and the optimistic version (28) of the two-stage equilibrium model (17)- (18) are equivalent in the sense that they have the same solution set. Proof. See Appendix A.3 for the proof. The result of this proposition is very interesting and justifies the two-stage SP model (19) , although the government becomes the only decision maker to medical supplies for an epidemic. The two-stage SP model (19) can be built directly from the view of optimality for the society and the government is the only decision maker. While the optimistic version (28) of the two-stage stochastic equilibrium model (17)-(18) is built by considering the competitions in the second stage for the dynamic distribution, and assume that the government is able to select an optimal decision for the whole society from the possible choices competed by the hospitals. The two models indeed provide the same solution sets. Thus we conclude that in medical supplies for an epidemic, a powerful and smart government can make decisions from the view of social optimality that are also acceptable for all hospitals in the sense that no hospital can decrease its own costs by unilaterally changing its strategy. 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 First of all, the nonsmooth term (M ijt (ξ)) + in the penalty term of (3) can be replaced by a more general nonsmooth function R((M ijt (ξ)) + ). For instance, the out-layer function R(ϕ) can be quadratic, or cubic, or piecewise linear as where 0 < σ 1 < σ 2 < 1 and 0 < δ 1 < δ 2 < δ 3 are given parameters. One can substitute (M ijt (ξ)) + by a new nonnegative variable θ ijt (ξ) in the penalty term of (3), and add the following constraints to the two-stage stochastic equilibrium model: for any i ∈ I, j ∈ J , and for a.e. ξ ∈ Ξ, The two-stage stochastic equilibrium model can be transformed to the smooth two-stage SP model. In case Ξ = {ξ 1 , . . . , ξ K }, this strategy will increase mnlK functional constraints and mnlK nonnegativity constraints to the original model. The strategy of this type is beneficial for mnlK of moderate size, but may increase computational cost dramatically when mnlK is huge, or facing more complex nonsmooth penalty functions such as the piecewise function R((M ijt (ξ)) + ) in (29) . The smoothing strategy we use in this paper can be easily extended to more complex nonsmooth penalty functions such as (29) , and only enrolls a smoothing parameter µ ∈ R instead. Secondly, it is very interesting to consider the case when the quantity of a type of non-reusable medical supply such as drugs delivered to hospital i on day t can be greater than the demands. Motivated by [2] , for such non-reusable medical supply j, we can modify the amount of unmet demands of hospital i for medical supply j on day t as 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Note that if M ijt (ξ) is defined in this way, in case that M ijt (ξ) < 0, i.e., the quantity of medical supply j that can be used on day t is more than the demands of hospital i, then the excess quantity will be delivered to the next day as part of the quantity that can be used. In case that M ijt (ξ) > 0, i.e., there exist unmet demands on day t, the unmet demands will be penalized each day if they are still not be covered in the following days. It is more reasonable to assign a time-variant penalty coefficient that increases with the delay time, e.g., where c ij is the unit penalty coefficient, t represents the day t, and l is the total number of days we consider. Here Q t l monotonically increases with t l . In [2] , Q( t l ) = ( t l ) 3 is employed. For the extensions mentioned above, the analysis also holds for the nonsmooth two-stage stochastic equilibrium model, its smooth approximation, as well as the corresponding two-stage SP and the two-stage SVI model. The PHM outlined in the next section is also applicable to solve the two-stage SVI models for extensions, with a slight modification of the function value and the Jacobian matrix of the function involved in the NCP. The discretized two-stage SVI (22) is a deterministic variational inequality (VI) which may be solved by any existing solvers. However, when K is large, it is more efficient to solve (22) using the PHM [14] that can exploit the two-stage structure. As shown in [14] , when (22) is of maximal monotone type, the PHM has solid convergence. The PHM has very clear structure and is easy to code. The efficient semismooth Newton method [42, 43] can be enrolled in the PHM for the two-stage SVI, and the efficiency of the PHM has been demonstrated in [31, 35] . Moreover, the PHM is attractive for dealing with further extensions of the proposed model, because it does not require the objective function to be linear or quadratic as some existing efficient solvers for the two-stage SP 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 do, including the "linprog" in Matlab and the Benders decomposition method [44, 45] . The PHM performs well when facing large number of scenarios, by using parallel computation. We use the notation y k = y(ξ k ), z k = z(ξ k ), s k = s(ξ k ), π yk = π y (ξ k ), π zk = π z (ξ k ), π sk = π s (ξ k ) and p k = p(ξ k ) for simplicity. Choose a step size r > 0 and a smoothing parameter µ > 0. Set ν = 0. Step 1. For k = 1, 2, . . . , K, solve the NCP using the semismooth Newton method and obtain a solution (x ν k ,λ ν k ,ŷ ν k ,ẑ ν k ,ŝ ν k ,π ν yk ,π ν zk ,π ν sk ), k = 1, 2, . . . , K. Step 2. For k = 1, 2, . . . , K, let Set ν = ν + 1, and go to Step 1. Note that the proximal term ensures the existence of a unique solution to the subproblem in (33) . The elements u ν k and v ν k have the role of Lagrange 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 multipliers corresponding to the nonanticipativity constraints for the vectors x and λ. We use the efficient semismooth Newton method [43] to solve the subproblems in (33) for k = 1, . . . , K. Denote For any ξ k , k = 1, . . . , K, let us denote We adopt a strategy for updating µ as similar to that used in the smoothing methods for solving constrained nonsmooth optimization [37, 38, 46, 47] . This strategy is helpful for accelerating the computational speed compared with that using a fixed µ, and is promising to get an approximate solution of the original nonsmooth two-stage SP. To be specific, we do the following procedures. • Choose a relatively large initial smoothing parameter µ 0 from the beginning, an acceptable level of the final smoothing parameter 0 < l µ < µ 0 , and constants γ > 0, δ µ ∈ (0, 1). 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 • For k ≥ 0, we use the iterates of the PHM (Algorithm 1) to approximately solve the two-stage SVI with µ ν . If η(µ ν ) < γµ ν and µ ν > l µ , that is, the two-stage SVI with µ ν is approximately solved but µ ν is not acceptable as the final smoothing parameter, we then set µ ν+1 = δ µ µ ν , otherwise, we set µ ν+1 = µ ν . For k = 1, . . . , K, define The implementation of the semismooth Newton method requires the Jacobian matrix J k of G(w k ). We give the form of the Jacobian matrix J k of G(w k ) below. Denote The Jacobian matrix of G(w k ) is 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 wherē The semismooth Newton method is efficient, since the Jacobian matrix J k is sparse and has special structure. In this section, we focus on the storage and dynamic distribution of medical 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Based on the volume and weight, shelf life, acquisition cost, warehouse rent, holding cost of each medical supply and the average number of years that of such epidemics occur, we estimate the unit reserve price of each medical supply for each hospital in Table 1 Table 2 lists the specific values of the above parameters. We make samples based on the daily hospital infection data from February 10th, 2020 to February 23rd, 2020, published by Wuhan Municipal Health Commission, as shown in Table 3 and plotted in Table 4 . In the tables, "MS" refers to medical supply. Unlike the other parameters that have clear rules for estimating, the penalty 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 coefficient c ij for unmet demands may follow different rules guided from the government to reflect fairness. We consider three different cases for the penalty coefficients described below. The specific vectors of penalty coefficients are listed in Table 5 . -Case 1: The vectors of penalty coefficients for the hospitals are the same. -Case 2: Two different vectors of penalty coefficients are adopted. One with lower values of components is for H1, H5 and H6 that mainly treated patients with mild infection, and the other is for the remaining four hospitals that mainly treated patients with severe infection. -Case 3: Each hospital is assigned a distinct vector of penalty coefficients. Not only the mild/severe degree of patients is addressed, but also the numbers of patients in hospitals are taken into consideration. Let "sn" be the number of scenarios. We randomly take sn scenarios within a band of a 10% fluctuation of the actual number of infected people as the possible scenarios of ξ with equal probability. We use the strategy in Section 5 for updating the smoothing parameter, where we set µ 0 = 20, l µ = 0.7, γ = 0.001, and α = 0.5 for all the tests. The computed solutions satisfy η(µ ν ) < 7 × 10 −4 and µ ν < l µ for all the different cases. The parameter r in the PHM in each experiment is set to be 1. 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Table 7 . To be specific, we use the strategy to assign an auxiliary vector θ ijt (ξ) as in (30) We compare the two-stage SVI model with the deterministic VI model for the three cases to evaluate their performances in addressing demand uncertainty. We use sn = 200 for the two-stage SVI model, and use the number of real infected people for the deterministic VI model. Table 8 lists the elapsed time. Table 9 lists the objective value (total cost), followed by specific costs including the acquisition and holding costs, the transportation costs, the government purchasing costs and the penalty costs, of the deterministic VI model and the two-stage SVI model, respectively, as well as the costs of substituting the solution of the deterministic VI model into the two-stage SVI model. The last column of Table 9 lists the percentage of improvement in the total cost achieved by using the two-stage SVI model instead of the deterministic VI model. The objective values of the deterministic VI model are approximately 3.64 × 10 9 CNY, 3.65 × 10 9 CNY, and 3.65 × 10 9 C-NY for the three cases, respectively. The objective values of the two-stage SVI model are increased to 3.67 × 10 9 CNY, 3.68 × 10 9 CNY, and 3.68 × 10 9 CNY 35 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 for the three cases, respectively. In reality, the demands of the deterministic VI model cannot be absolutely reliable. We will suffer when sticking to the nominal solution obtained by the deterministic VI model for all possible scenarios, by noting that the objective values of the two-stage SVI models at the solution of the deterministic VI models are much higher than those of using the optimal solutions of the two-stage SVI models for all the three cases. Below we focus on analyzing the numerical results obtained by our proposed two-stage SVI model and the PHM with updating strategy for µ. We take sn = 20. We compare the optimal values in the first stage from no storage to sufficient upper limits of storage in all the three cases. The upper limit x j max increases from u 1 to u 10 as shown in Table 10 , which is essentially determined by the budgets. Because inspection and testing articles cannot be stored before an epidemic, the upper limit x 2 max is always set to be 0. We set the upper limit of storage capacity x ij max here sufficiently large. 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 The objective values of the first stage corresponding to u 1 , ..., u 10 are plotted in Figure 3 . For each case, the highest total cost is obtained at the level of CNY and 591,774,405 CNY for the three cases, respectively. If we further increase the budget for storage, we will get no further improvement on saving the total cost. We fix x j max at the level of u 2 , and record the computed solutions in Table 11 for the storage decision in the first stage of our proposed two-stage SVI model. We find that the sum of all the hospitals' reserves has reached the upper limit for each of the three cases. This indicates that if the budgets are not sufficient, storing medical supplies as much as possible before an epidemic is a good decision to save money and to meet the medical demands later on. We can see from Table 11 that when the penalty coefficients are completely 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 the storage amounts of medical supplies pre-epidemic in order to minimize the total cost under the given budgets, by considering both the unit reserve prices and the penalty coefficients. Because the housing prices of different geographical locations are different, we set the hospitals' unit reserve prices different in Table 1 . In order to understand the effect of unit reserve prices on the solution, we do numerical experiments with the same unit reserve prices and the same penalty coefficients. Table 12 lists the optimal solution of storage for this case. By comparing with the solution of Case 1 in Table 11 , we can find that the solution in Table 12 tends to be balance when the unit reserve prices are the same. In this subsection, we fix x j max at the level of u 2 for the two-stage SVI model, and concentrate on the dynamic distribution in the second stage. The dynamic distribution is determined after the realization of scenario is grasped. Number of test reusable medical supplies, the distribution curves of reusable medical supplies for the three cases are basically the same. That is, in optimal solutions, the reusable medical supplies are offered as early as possible. For each hospital i and each medical supply j, the average amount Aveγ ij (ξ) of medical supplies obtained per person relates closely to the fairness, which can be computed as follows. We show in Figure 12 the average amounts of non-reusable medical sup- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 amounts of non-reusable medical supplies obtained per person in hospitals with lower penalty coefficients are relatively small. The three cases actually reflect the different attitudes to fairness. In Case 1, we do not distinguish hospitals and set the penalty coefficients all the same. The results indeed reflect the fairness in the sense of near "equitability". That is, all hospitals are serviced as approximately indistinguishable entities. We believe that the small differences of unit reserve prices among hospitals lead to the minor differences of the average amounts of medical supplies obtained. In Case 2, we divide the hospitals into two groups that treated patients with mild and severe infection, respectively. The four hospitals that treated severe patients receive similar higher average amounts of medical supplies, which indicates that they get priorities of medical supplies. The hospitals that treated mild patients receive similar lower average amounts of medical supplies. This decision also reflects the fairness in the sense of "balance" that hospitals are serviced with medical supplies regarding their severe/mild infections of patients. In Case 3, the results show that the higher the penalty coefficients are, the more the amounts of the medical supplies obtain. This is also the fairness in the sense of "balance" regarding not only the difference of mild/severe infections but also the total number of infected people in the hospitals as we desire. While in contrast, reusable medical supplies almost have no shortages. Hence they basically have met the demands of all the hospitals throughout the time horizons for the three cases as shown in Figure 13 . We take into account all the medical supplies together and compare the average cost per patient over the 14 days of each hospital for the three cases, as shown in Figure 14 . For hospital i, the average cost can be computed by the formula below. It can be seen that when the penalty coefficients are the same (Case 1), the average costs per patient of the 7 hospitals are very close; when the penalty coefficients are different (Case 2 and Case 3), the average costs per patient 48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 Finally, we do preliminary numerical experiments for the extension of the model in Subsection 4.2 that the quantities of drugs delivered to hospital i on day t can be greater than the demands. We use formula (31) for the amount of unmet demands, and assign Q( t l ) = ( t l ) 3 in (32) for medical supply 1 -drugs. If we still use the original y 1 max = 10000 in Table 2 , there is no quantity of drugs delivered to hospital i on day t that is greater than the demand at the computed solution for each case. We then increase y 1 max = 13000. Figure 15 shows the obtained amounts of drugs in each hospital for 14 days for the three cases. Due to the sufficient supplies of drugs, almost all the hospitals choose to purchase the excess quantity at a low price in advance for future use, and consequently the distribution curves of drugs for the three cases are basically the same. Figure 16 shows the average amounts of drugs obtained per person in each hospital for the three cases, which are approximately the same. This is reasonable since the demands of all the hospitals throughout the time horizons for the three cases have been met. 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 Numerical results show that our model is flexible to obtain a solution of desired "fairness", and our solution for the storage and dynamic distribution of medical supplies in Wuhan can clearly save money. In the future, we will consider more hospitals involved in the second stage. With the increasement of hospitals, the dimensions of decision vectors will become much higher, which may lead to computational inefficiency when using the semismooth Newton method in the PHM. It is worthwhile to study whether there are strategies, and/or new methods to solve large-scale models efficiently. In addition, we would like to add the decision variables for locations of warehouses. This is an important issue to be addressed in the first stage, but usually introduces integer variables in the model . The computation would be 52 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 more challenging. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 below min x,y(ξ k ),z(ξ k ),s(ξ k ) a T x + e (m) TF (y(ξ k ), z(ξ k ), s(ξ k ), µ) x, y(ξ k ), z(ξ k ), s(ξ k ) ≥ 0. (A. 2) The monotonicity of the two-stage SVI, together with the fact the involved functions defined for the two-stage SVI are continuous, yields that (22) Proof. Note that the right-hand side vectors in (5)-(7) are positive vectors. There may exist components in the right-hand side vectors (4) to be zero, since some medical supplies are not available before an epidemic occurs. Let J 0 = {j ∈ J : x j max = 0}. For any vector u yzs := y(ξ k ) T , z(ξ k ) T , s(ξ k ) T , k = 1, . . . , K T ∈ S 2 (x), it is obvious from constraint (4) that x ij = 0, ∀i ∈ I, j ∈ J 0 , (A. 3) and consequently from constraint (8) s ijt (ξ k ) = 0, ∀t ∈ T , j ∈ J 0 , k = 1, . . . , K. Denotex the subvector of x, ands(ξ k ) the subvector of s(ξ k ), andπ(ξ k ) the subvector ofπ s (ξ k ), by deleting all components that correspond to j ∈ J 0 . We obtain a reduced two-stage stochastic equilibrium problem, by deleting the components of x and s(ξ k ) in the objective functions (2) and (3) that correspond to j ∈ J 0 , the constraints in (4), (8) , and (9), and the constraints 54 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 s ijt (ξ k ) ≥ 0 in (10) for j ∈ J 0 . LetS 2 (x) be the solution set of the reduced second-stage equilibrium problem. The reduced two-stage equilibrium problem is equivalent to the reduced NCP by deleting the complementarity conditions in (26) that correspond to the variables of s(ξ k ) andπ s (ξ k ) for j ∈ J 0 , the constraints in (4), (8) , and (9), and the constraints s ijt (ξ k ) ≥ 0 in (10) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 Because the objective value at an optimal solution is no more than the objective value at any feasible solution, we get from (28) and (19) that τ k e (m) TF (ŷ(ξ k ),ẑ(ξ k ),ŝ(ξ k ), µ) τ k e (m) TF (y * (ξ k ), z * (ξ k ), s * (ξ k ), µ) τ k e (m) TF (ŷ(ξ k ),ẑ(ξ k ),ŝ(ξ k ), µ). Consequently, the solution (x * , u * yzs ) of the two-stage SP model (19) is also a solution of the optimistic version (28) of the two-stage stochastic equilibrium model (17)- (18) , and conversely the solution (x,û yzs ) of the optimistic version (28) is also a solution of the two-stage SP model (19) . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 Pre-positioning and dynamic delivery planning for short-term response following a natural disaster Multi-period dynamic distributionally robust pre-positioning of emergency supplies under demand uncertainty A multi-objective evolutionary algorithm for multi-period dynamic emergency resource scheduling problems Dynamic resource allocation to support oil spill response planning for energy exploration in the arctic A dynamic logistics coordination model for evacuation and support in disaster response activities A dynamic multi-objective location-routing model for relief logistic planning 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programming Two-stage stochastic variational inequality arising from stochastic programming Two-stage non-cooperative games with risk-averse players Regularized two-stage stochastic variational inequalities for Cournot-Nash equilibrium under uncertainty Two-stage quadratic games under uncertainty and their solution by proressive hedging algorithms Two-stage stochastic variational inequalities: theory, algorithms and applications Smoothing projected gradient method and its application to stochastic linear complementarity problems Smoothing methods for nonsmooth, nonconvex minimization Smooth minimization of non-smooth functions Introductory Lectures on Convex Optimization: A Basic Course An overview of bilevel optimization A nonsmooth version of Newton's method A semismooth equation approach to the solution of nonlinear complementarity problems Partitioning procedures for solving mixed-variables programming problems Benders Decomposition Smoothing SQP methods for solving degenerate nonsmooth constrained optimization problems with applications to bilevel programs A smoothing active set method for linearly constrained non-Lipschitz nonconvex optimization Variational Analysis Finite-Dimensional Variational Inequalities and Complementarity Problems The authors would like to thank the associate editor and the anonymous ref- Proof. According to Theorem 3.5 of [13] , the monotonicity of the SVI (22) is equivalent to the monotonicity of the VIfor each k = 1, . . . , K. For each k, the above VI is monotone, since it is just the KKT system of the convex programming for each scenario ξ k relating to (21) Appendix A.3. Proof of Proposition 3Proof. We denote for simplicity u yzs := y(ξ k ) T , z(ξ k ) T , s(ξ k ) T , k = 1, . . . , K T .Let (x * , u * yzs ) be a solution of the two-stage SP model (19) . Then there exist the vectors of Lagrange multipliers λ * , π * y (ξ k ), π * z (ξ k ), and π * s (ξ k ) for k = 1, . . . , K, together with x * and u * yzs , that satisfy the KKT system (22) . Thus u * yzs , together with the vectors of Lagrange multiplierŝfor k = 1, . . . , K, satisfies the system (26) with x = x * . Hence u * yzs ∈ S 2 (x * ), and (x * , u * yzs ) is a feasible solution of the optimistic version (28) of the smooth two-stage equilibrium model (17)- (18) .Let (x,û yzs ) be a solution of the optimistic version (28) of the two-stage equilibrium model (17)- (18) , which is clear to be a feasible solution of the twostage SP model (19) .