American Institute of Mathematical Sciences

Complex intuitionistic fuzzy sets (CIFSs), characterized by complex-valued grades of membership and non-membership, are a generalization of standard intuitionistic fuzzy (IF) sets that better speak to time-periodic issues and handle two-dimensional data in a solitary set. Under this environment, in this article, various mean-type operators, namely complex IF Bonferroni means (CIFBM) and complex IF weighted Bonferroni mean (CIFWBM) are presented along with their properties and numerous particular cases of CIFBM are discussed. Further, using the presented operators a decision-making approach is developed and is illustrated with the help of a practical example. Also, the reliability of the developed methodology is investigated with the aid of validity test criteria and the example results are compared with prevailing methods based on operators.

There is a wide consensus that the shadow prices of certain resources in an economic system are equal to Lagrange multipliers. However, this is misleading with respect to multiple Lagrange multipliers. In this paper, we propose a new type of Lagrange multiplier, the weighted minimum norm Lagrange multiplier, which is a type of shadow price. An attractive aspect of this type of Lagrange multiplier is that it conveys the sensitivity information when resources are required to be proportionally input. To compute the weighted minimum norm Lagrange multiplier, we propose two algorithms. One is the penalty function method with numeric stability, and the other is the accelerated gradient method with fewer arithmetic operations and a convergence rate of \begin{document}$ O(\frac{1}{k^2}) $\end{document}. Furthermore, we propose a two-phase procedure to compute a particular subset of shadow prices that belongs to the set of bounded Lagrange multipliers. This subset is particularly attractive since all its elements are computable shadow prices. We report the numerical results for randomly generated problems.

The paper proposes a novel class of quadratically constrained convex reformulations (QCCR) for semi-continuous quadratic programming. We first propose the class of QCCR for the studied problem. Next, we discuss how to polynomially find the best reformulation corresponding with the tightest continuous bound within this class. The properties of the proposed QCCR are then studied. Finally, preliminary computational experiments are conducted to illustrate the effectiveness of the proposed approach.

This paper considers the principal component analysis when the covariance matrix of the input vectors drops rank. This case sometimes happens when the total number of the input vectors is very limited. First, it is found that the eigen decomposition of the covariance matrix is not uniquely defined. This implies that different transform matrices could be obtained for performing the principal component analysis. Hence, the generalized form of the eigen decomposition of the covariance matrix is given. Also, it is found that the matrix with its columns being the eigenvectors of the covariance matrix is not necessary to be unitary. This implies that the transform for performing the principal component analysis may not be energy preserved. To address this issue, the necessary and sufficient condition for the matrix with its columns being the eigenvectors of the covariance matrix to be unitary is derived. Moreover, since the design of the unitary transform matrix for performing the principal component analysis is usually formulated as an optimization problem, the necessary and sufficient condition for the first order derivative of the Lagrange function to be equal to the zero vector is derived. In fact, the unitary matrix with its columns being the eigenvectors of the covariance matrix is only a particular case of the condition. Furthermore, the necessary and sufficient condition for the second order derivative of the Lagrange function to be a positive definite function is derived. It is found that the unitary matrix with its columns being the eigenvectors of the covariance matrix does not satisfy this condition. Computer numerical simulation results are given to valid the results.

In this paper we propose a robust and efficient primal-dual interior-point method for a nonlinear ill-conditioned problem with associated errors which are arising in the unfolding procedure for neutron energy spectrum from multiple activation foils. Based on the maximum entropy principle and Boltzmann's entropy formula, the discrete form of the unfolding problem is equivalent to computing the analytic center of the polyhedral set \begin{document}$ P = \{x \in R^n \mid Ax = b, x \ge 0\} $\end{document}, where the matrix \begin{document}$ A \in R^{m\times n} $\end{document} is ill-conditioned, and both \begin{document}$ A $\end{document} and \begin{document}$ b $\end{document} are inaccurate. By some derivations, we find a new regularization method to reformulate the problem into a well-conditioned problem which can also reduce the impact of errors in \begin{document}$ A $\end{document} and \begin{document}$ b $\end{document}. Then based on the primal-dual interior-point methods for linear programming, we propose a hybrid algorithm for this ill-conditioned problem with errors. Numerical results on a set of ill-conditioned problems for academic purposes and two practical data sets for unfolding the neutron energy spectrum are presented to demonstrate the effectiveness and robustness of the proposed method.

With regard to environmental pressures and economic benefits, some original construction equipment manufacturers, have focused on collecting and recovering construction machinery at the end of their life. The present study aimed to focus on Sustainable closed-loop supply chain network optimization for construction machinery recovering. To this purpose, different recovery options such as remanufacturing, recycling and reusing were implemented. A mixed integer linear programming model (MILP) including three objective functions was proposed in this regard. Based on the model, all three dimensions of sustainability including economic, environmental, and social dimensions were considered and could successfully determine the optimal values of the flow of used products, remanufactured products, recycled parts, re-usable parts. In order to demonstrate the applicability of the proposed model, a numerical example was used with the help of GAMS software to obtain the supply chain structure with the lowest cost and reduce the pollution caused by CO₂. Finally, the model could maximize fixed and variable job opportunities.

In this paper, we consider the quantitative stability of a class of risk-averse multistage stochastic programs, whose objective functions are defined by multi-period \begin{document}$ p $\end{document}th order lower partial moments (LPM) with given targets, and their distributionally robust counterparts. We first derive the upper bounds of feasible solutions as preliminaries. Then, by employing calm modifications, the quantitative stability results are obtained under a special measurable perturbation of stochastic process, which extend the present results under risk-neutral cases to risk-averse ones. Moreover, we recast the risk-averse model by probability measures of stochastic process, and obtain new quantitative stability estimations on the basis of proper probability metrics under the general perturbation of stochastic process. Finally, motivated by the availability of only partial information about probability measures, we further consider the distributionally robust counterpart of our recasting model, and establish the discrepancy of optimal values with respect to the perturbation of ambiguity sets.

We propose a novel method for constructing probabilistic robust disturbance rejection control for uncertain systems in which a scenario optimization method is used to deal with the nonlinear and unbounded uncertainties. For anti-disturbance, a reduced order disturbance observer is considered and a state-feedback controller is designed. Sufficient conditions are presented to ensure that the resulting closed-loop system is stable and a prescribed \begin{document}$ H_{\infty} $\end{document} performance index is satisfied. A numerical example is presented to illustrate the effectiveness of the techniques proposed and analyzed.

In reality, a contractor may implement multiple projects simultaneously and in such an environment, how to achieve a positive balance between cash outflow and inflow by scheduling is an important problem for the contractor has to tackle. For this fact, this paper investigates a resource-constrained multi-project scheduling problem with the objective of minimizing the contractor's maximal cash flow gap under the constraint of a project deadline and renewable resource. In the paper, we construct a non-linear integer programming optimization model for the studied problem at first. Then, for the NP-hardness of the problem, we design three metaheuristic algorithms to solve the model: tabu search (TS), simulated annealing (SA), and an algorithm comprising both TS and SA (SA-TS). Finally, we conduct a computational experiment on a data set coming from existing literature to evaluate the performance of the developed algorithms and analyze the effects of key parameters on the objective function. Based on the computational results, the following conclusions are drawn: Among the designed algorithms, the SA-TS with an improvement measure is the most promising for solving the problem under study. Some parameters may exert an important effect on the contractor's maximal cash flow gap.

Carbon emission reduction is regarded as an effective way to protect the environment, which requires a large amount of capital. Thus, for a remanufacturing firm with limited initial capital, trade credits act as an effective financing method in supporting production and emission reductions. In this study, under the cap-and-trade and government's subsidy policies, a joint decision on recycling, remanufacturing and emission reduction by a financially constrained remanufacturer with considering deferred payment to a third-party recycler is analyzed. On the basis, optimization models are established to derive the optimal recycling quantity, carbon reduction rate and government subsidy rate by using a backward induction. Furthermore, an analytical comparison is provided between the cases of base model, carbon abatement investment model and deferred payment model. Numerical experiment results indicate that the remanufacturer can always make use of the investment option to further decrease its carbon emissions and gain more profit. We also find that deferred payment can effectively mitigate carbon emissions only when the degree of emission efforts is more than a certain critical value, and it also plays a positive role in the third-party recycler's revenue, especially for the case with higher initial capital. Some other managerial implications are further discussed.

Since agricultural cooperatives have developed rapidly under the farmland transfer policy in China, they play an important role in the new operation pattern for China's fresh agricultural product supply chains. To enhance current agricultural supply chains' stability, we consider a three-level (farmer-cooperative-retailer) fresh agricultural product supply chain, conduct quantitative analysis of the impact of the quantity flexibility contract, and compare the impact of the relational contract with that of the quantity flexibility contract on the freshness and the profit. Our results show that a suitable relational contract can improve the freshness and increase the profit of the three-level supply chain, but cannot fully guarantee its stability. Furthermore, the government's subsidy policy can improve the relational contract stability of the fresh agricultural product supply chain by providing the protection price contract mechanism of the agricultural product and the cold chain facility subsidy contract.

In this paper, we employ an efficient numerical method to solve transport equations with given boundary and initial conditions. By the weighted-orthogonal Chebyshev polynomials, we design the corresponding basis functions for spatial variables, which guarantee the stiff matrix is sparse, for the spectral collocation methods. Combining with direct algebraic algorithms for the sparse discretized formula, we solve the equivalent scheme to get the numerical solutions with high accuracy. This collocation methods can be used to solve other kinds of models with limited computational costs, especially for the nonlinear partial differential equations. Some numerical results are listed to illustrate the high accuracy of this numerical method.

The development of the Internet has dramatically changed firms' business models. Companies can now use both virtual and physical channels to enhance their competitiveness and profitability. In addition, bundling is a commonly used promotion strategy, although managers should consider the characteristics of the candidate bundled products. This study proposes a two-stage game theoretic model, in which a manufacturer may start an online channel along with an existing physical one which is operated by a dealer, i.e., a bricks-and-clicks approach, to examine the bundling and pricing strategy when selling two products with different network externalities. In the first stage, the manufacturer offers the products to the dealer, who may sell the two products individually or in a bundle to customers. In the second stage, and with the aim of expanding market share, the manufacturer may consider starting an online channel to integrate with the existing physical channel. We consider four cases, in which the manufacturer and dealer may sell the two products either individually or bundled in the two channels, in order to obtain the corresponding optimal pricing strategies with the aim of maximizing their profits. We also perform a numerical analysis to investigate the effects that network externality has on the bundling strategies and profits of the two channels. The results indicate that the bricks-and-clicks business model benefits both the manufacturer and dealer, and their profits would increase as network externality increases. In particular, when the network externalities of the two products are both high, a mixed strategy, which sells the two products in a bundle in the online channel and individually in the physical channel, should be adopted.

The multiple-sets split equality problem is an extended form of the split feasibility problem. It has a wide range of applications in image reconstruction, signal processing, computed tomography, etc. In this paper, we propose a relaxed successive projection algorithm to solve the multiple-sets split equality problem which does not need the prior knowledge of the operator norms, and prove the strong convergence of the algorithm. The numerical examples indicate that the algorithm has good feasibility and effectiveness by comparing with other algorithm.

The paper deals with the application of the survival theory in economic systems. Theory and methodology of survival is used to evaluate fiscal policy. The survival of the system reduces to a problem of maximizing a radius of a cube inscribed into a polyhedral set so-called the target-oriented purpose [1-5]. We show that the survival theory can be applied to the government fiscal policy optimizing a taxation system. Numerical simulations were conducted using Mongolian statistical data for 2015.

We investigate a principal-agent model featured with unknown agent ability. Under the exponential utilities, the necessary and sufficient conditions of the incentive contract are derived by utilizing the martingale and variational methods, and the solutions of the optimal contracts are obtained by using the stochastic maximum principle. The ability uncertainty reduces the principal's ability of incentive provision. It is shown that as time goes by, the information about the ability accumulates, giving the agent less space for belief manipulation, and incentive provision will become easier. Namely, as the contractual time tends to infinity (long-term), the agent ability is revealed completely, the ability uncertainty disappears, and the optimal contracts under known and unknown ability become identical.

The problem of maximizing a given set function with a cardinality constraint has widespread applications. A number of algorithms have been provided to solve the maximization problem when the set function is monotone and submodular. However, reality-based set functions may not be submodular and may involve large-scale and noisy data sets. In this paper, we present the Stochastic-Lazier-Greedy Algorithm (SLG) to solve the corresponding non-submodular maximization problem and offer a performance guarantee of the algorithm. The guarantee is related to a submodularity ratio, which characterizes the closeness of to submodularity. Our algorithm also can be viewed as an extension of several previous greedy algorithms.

Although intuitionistic fuzzy preference relations have become powerful techniques to express the decision makers' preference information over alternatives or criteria in group decision making, some limitations of them are pointed out in this paper, then they are overcame by developed the group decision making with Pythagorean fuzzy preference relations (PFPRs). Specially, we provide a partial order on the set of all the PFPRs, based on which, a deviation measure is defined. Then, we check and reach the acceptably multiplicative consistency and consensus of PFPRs associated with the partial order and mathematical programming. Concretely, acceptably multiplicative consistent PRPRs are defined by the deviation between a given PFPR and a multiplicative consistent PFPR constructed by a normal Pythagorean fuzzy priority vector. Then acceptable consensus of a collection of PFPRs is defined by the deviation of each PFPR and the aggregated result from symmetrical Pythagorean fuzzy aggregation operators. Based on which, a method which can simultaneously modify the unacceptable consistency and consensus of PFPRs in a stepwise way is provided. Particularly, we also prove that the collective PFPR obtained by aggregating several individual acceptably consistent PFPRs with various symmetric aggregation operators is still acceptably consistent. Then, a procedure is provided to solve group decision making with PRPRs and a numerical example is given to illustrate the effectiveness of our method.

In the dual risk model, we study the periodic dividend problem with a non-exponential discount function which results in a time-inconsistent control problem. Viewing it within the game theoretic framework, we extend the Hamilton-Jacobi-Bellman (HJB) system of equations from the fixed terminal to the time of ruin and derive the verification theorem, and we generalize the theory of classical optimal periodic dividend. Under two special non-exponential discount functions, we obtain the closed-form expressions of equilibrium strategy and the corresponding equilibrium value function in a compound Poisson dual model. Finally, some numerical examples are presented to illustrate the impact of some parameters.

In this paper, we propose a new combined scalarization method of multi-objective optimization problems by using the surplus variables and the generalized Tchebycheff norm and then use it to obtain some equivalent scalarization characterizations of (weakly, strictly, properly) efficient solutions by adjusting the range of parameters. These scalarization results do not need any convexity assumption conditions of objective functions. Furthermore, we establish some scalarization results of approximate solutions by means of the method. Moreover, we also present some examples to illustrate the main results.

This article provides new developments in characterizing the class of regime-switching exponential affine interest rate processes in the context of pricing a zero-coupon bond. A finite-state Markov chain in continuous time dictates the random switching of time-dependent parameters of such processes. We present exact and approximate bond pricing formulas by solving a system of partial differential equations and minimizing an error functional. The bond price expression exhibits a representation that shows how it is explicitly impacted by the rate matrix and the time-dependent coefficient functions of the short rate models. We validate the bond pricing formulas numerically by examining a regime-switching Vasicek model.

A lookback option is an exotic option that allows investors to look back at the underlying prices occurring over the life of the option, and to exercise the right at assets optimal point. This paper proposes a mean-reverting stock model to investigate the lookback option in an uncertain environment. The lookback call and put options pricing formulas of the stock model are derived, and the corresponding numerical algorithms are designed to compute the prices of these two options.

Alternating direction methods of multipliers (ADMM) have been well studied and effectively used in various application fields. The classical ADMM must solve two subproblems exactly at each iteration. To overcome the difficulty of computing the exact solution of the subproblems, some proximal terms are added to the subproblems. Recently, {{a special proximal ADMM has been studied}} whose regularized matrix in the proximal term is generated by the BFGS update (or limited memory BFGS) at every iteration for a structured quadratic optimization problem. {{The numerical experiments also showed}} that the numbers of iterations were almost same as those by the exact ADMM. In this paper, we propose such a proximal ADMM for more general convex optimization problems, and extend the proximal term by the Broyden family update. We also show the convergence of the proposed method under standard assumptions.

We propose a parallel iterative scheme with viscosity approximation method which converges strongly to a solution of the multiple-set split equality common fixed point problem for quasi-pseudocontractive mappings in real Hilbert spaces. We also give an application of our result to approximation of minimization problem from intensity-modulated radiation therapy. Finally, we present numerical examples to demonstrate the behaviour of our algorithm. This result improves and generalizes many existing results in literature in this direction.

In this paper, a differentiable vector optimization problem with the multiple interval-valued objective function and with both inequality and equality constraints is considered. The Karush-Kuhn-Tucker necessary optimality conditions are established for such a differentiable interval-valued multiobjective programming problem. Further, a new approach, called \begin{document}$ F $\end{document}-objective function method, is introduced for solving the considered differentiable vector optimization problem with the multiple interval-valued objective function. In this method, its associated vector optimization problem with the multiple interval-valued \begin{document}$ F $\end{document}-objective function is constructed. Their equivalence is established under \begin{document}$ F $\end{document}-convexity assumptions. It is shown that the introduced approach can be used to solve a nonlinear nonconvex interval-valued optimization problem. By using the introduced approximation method, it is also presented in some cases that a nonlinear nonconvex interval-valued optimization problem can be solved by the help of methods for solving linear interval-valued optimization problems.

The problem of the optimal location-allocation of processing factory and distribution center for supply chain networks under uncertain transportation cost and customer demand are studied. We establish a two-stage mean-risk stochastic 0-1 mixed integer optimization model, by considering the uncertainty and the risk measure of the supply chain. Given the complexity of the model this paper proposes a modified hybrid binary particle swarm optimization algorithm (MHB-PSO) to solve the resulting model, yielding the optimal location and maximal expected return of the supply chain simultaneously. A case study of a bread supply chain in Shanghai is then presented to investigate the specific influence of uncertainties on the food factory and distribution center location. Moreover, we compare the MHB-PSO with hybrid particle swarm optimization algorithm and hybrid genetic algorithm, to validate the proposed algorithm based on the computational time and the convergence rate.

We are concerned with the stability of the ground state for the Schrödinger-Poisson equation

\begin{document}$ i\frac{\partial\psi}{\partial t}+\triangle\psi-(|x|^{-1}\ast|\psi|^2)\psi+|\psi|^{p-1}\psi = 0,\quad x\in \mathbb{R}^3. $\end{document}

If \begin{document}$ 2<p<\frac{7}{3} $\end{document} and the frequency is sufficiently large, we show that the ground state is orbitally stable.

We study flowshop scheduling problems with respect to slack due window assignments, which are operations in which jobs are assigned an individual due window. We combine learning effect and controllable processing times, in which the flowshop has a two-machine no-wait setup. The goal is to determine job sequence, slack due window based on common flow allowance, due window size, and resource allocation. We provide a bicriteria analysis for the scheduling and resource consumption costs. We show that the two costs can be solved in polynomial time utilizing three different combinations.

As manufacturers may engage in both direct sale and wholesale, the channel conflict between manufacturer and retailer becomes inevitable. This paper considers a dual-channel supply chain in which a retailer sells the product through store channel with sales effort while the manufacturer holds a direct channel and may provide an incentive measure to share the cost of sales effort. To meet social responsibility, a penalty on the total resource consumed is imposed on the manufacturer. We present a manufacturer-led decentralized model in which both members maximize individual profit, and then derive the corresponding optimal direct/store price and wholesale price. The dual-channel supply chain model without sales effort policy is also considered so as to explain the effects of sales effort policy and sharing cost measure on both parties. Special properties are presented to show (ⅰ) the influence of retailer's sales effort and manufacturer's sharing cost on the optimal strategies; (ⅱ) the resource-utilized penalty on the optimal decisions. Finally, numerical experiments are conducted to highlight the influence of various parameters on optimal solutions. We find that if the market response to retailer's sales effort is strong or the manufacturer's sharing portion of sales effort cost is increased, the retailer's profit and store selling price increase while the manufacturer's profit decreases and the direct sale and wholesale prices do not change. We also show that if the consumer's value on direct channel exceeds a threshold, the manufacturer's profit will be greater than that of the retailer. Moreover, if the market response to retailer's sales effort is strong, manufacturer's profit will be lesser than retailer's profit.

In reality, supply chain member may apply for loan from bank when he\begin{document}$ \backslash $\end{document}she is capital-constrained, or may deposit idle capital to bank when he\begin{document}$ \backslash $\end{document}she is well-funded. This study focuses on the Stackelberg pricing policy considering bank's deposit and loan based on delay payment scheme in a dyadic capital-constrained supply chain. First, the market demand is given, and then the profit functions of supply chain members are built. According to Stackelberg game, the four pricing models are constructed for four scenarios. By solving models, optimal pricing policies of supply chain members for each scenario can be determined. Finally, impacts of the interest rate for fixed deposit by installments, deposit rate and loan rate on optimal pricing policies are analyzed. The research results show that, in manufacturer capital-constrained situation, these rates can affect optimal pricing policies and profits; in retailer capital-constrained situation, deposit rate and loan rate have no effect on them, but the interest rate for fixed deposit by installments can still affect them. Our study provides a feasible way for supply chain members in pricing decision considering bank's deposit and loan based on delay payment scheme in a dyadic capital-constrained supply chain, and contributes to the theoretical research on the capital-constrained supply chain management and the management practice for capital-constrained supply chain members with bank' deposit and loan.

In the analyses on economic growth factors, researchers generally use the production function model to calculate the contribution rates of influencing factors to economic growth. The paper proposes a new modified VES production function model. As for the model's parameter estimation, the conventional optimization methods are complicated, generally require information like the gradient of objective function, and have the poor convergence rate and precision. The paper gives a modern intelligent algorithm, i.e., the cuckoo search algorithm, which has the strong robustness, can be realized easily, has the fast convergence rate and can be used flexibly. To enhance the convergence rate and precision, the paper improves the conventional cuckoo search algorithm. Using the new model, the paper gives a method calculating the contribution rates of economic growth influencing factors scientifically. Finally, the paper calculates the contribution rates of influencing factors to economic growth in Shanghai City, China.

Quality competition and risk aversion have become more and more common in today's many industries, making it a challenge to supply chain management and coordination. This paper considers a vendor-managed inventory (VMI) supply chain comprising two risk-averse manufacturers who sell their competing products through a common retailer. Market demand shared by each manufacturer is dependent on the quality level of its own product as well as on the competitor's product quality. The Conditional Value-at-Risk (CVaR) criterion is employed to formulate the risk aversion of manufacturers. This study first develops basic models without coordination mechanism and analyzes the effect of the quality sensitivity, competition intensity, risk aversion degree and cost coefficient of quality improvement on equilibrium decisions and supply chain efficiency. Further, a combined contract composed of option and cost-sharing is proposed to investigate the supply chain coordination issue. The results reveal that the combined contract can coordinate the supply chain and achieve a win-win outcome only when the manufacturers are low in risk aversion, and the system-wide profit of the supply chain can be allocated arbitrarily only by the option price. Also, this research examines the effect of the quality sensitivity, competition intensity, risk aversion degree and cost coefficient of quality improvement on the feasible region of option price.

Batch arrival and batch service queueing systems are of importance in the context of telecommunication networks. None of the work reported so far consider the dependence of consecutive arrival and service batches. Batch Markovian Arrival Process(\begin{document}$ BMAP $\end{document}) and Batch Markovian Service Process (\begin{document}$ BMSP $\end{document}) take care of the dependence between successive inter-arrival and service times, respectively. However in real life situations dependence between consecutive arrival and service batch sizes also play an important role. This is to regulate the workload of the server in the context of service and to restrict the arrival batch size when the flow is from the same source. In this paper we study a queueing system with Markov dependent arrival and service batch sizes. The arrival and service batch sizes are assumed to be finite. Further, successive inter-arrival and service time durations are also assumed to be correlated. Specifically, we consider a \begin{document}$ BMAP/BMSP/1 $\end{document} queue with Markov dependent arrival and Markov dependent service batch sizes. The stability of the system is investigated. The steady state probability vectors of the system state and some important performance measures are computed. The Laplace-Stieltjes transform of waiting time and idle time of the server are obtained. Some numerical examples are provided.

Dynamic network flow problems have wide applications in evacuation planning. From a given subset of arcs in a directed network, choosing the suitable arcs for facility location with a given objective is very important in the optimization of flow in emergency cases. Because of the decrease in capacity of an arc by placing a facility in it, there may be a reduction in the maximum flow or increase in the quickest time. In this work, we consider a problem of identifying the optimal facility locations so that the increase in the quickest time is minimum. Introducing the quickest FlowLoc problem, we give strongly polynomial time algorithms to solve the single facility case. Realizing NP-hardness of the multi-facility case, we develop a mixed integer programming formulation of it and propose two polynomial time heuristics for its solution. Because of the growing concerns of arc reversals in evacuation planning, we introduce the quickest ContraFlowLoc problem and present exact algorithms to solve the single-facility case and heuristics to solve the multi-facility case, with polynomial time complexity. The solutions thus obtained here are practically important, particularly in evacuation planning, to systematize traffic flow with facility allocation minimizing evacuation time.

In this paper, we study the optimal reinsurance design with default risk by minimizing the VaR (value at risk) of the reinsurer's total risk exposure. The optimal reinsurance treaty is provided. When the reinsurance premium principle is specified to the expected value and exponential premium principles, the explicit expressions for the optimal reinsurance treaties are given, respectively.