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In this paper, we study the portfolio selection problem considering the uncertain investment time horizon. We employ the Period Value at Risk (PVaR) to characterise the risk in the special cases and establish a fundamental mixed integer linear programming (MILP) model to obtain the optimal portfolio solutions. Identifying that the symmetric property of PVaR can significantly reduce the computation burden of the CPLEX solver, an enhanced MILP model is proposed. To verify the quality of obtained solutions, we also develop fast lower and upper bound estimation schemes respectively. Using the real-world data sets from NYSE and NASDAQ stock markets, numerical results are provided to show the efficiency of the enhanced model. [ABSTRACT FROM AUTHOR]

Quantity discount is a frequently adopted scheme that has not been explicitly investigated in logistics service procurement auctions. This paper focuses on a revised winner determination problem under quantity discounts and demand uncertainty for a fourth-party logistics (4PL) provider in a combinatorial reverse auction. To characterize our research problem, a two-stage stochastic nonlinear programming model is constructed. Inspired by the idea of sample average approximation (SAA), the nonlinear model is reformulated as a deterministic mixed integer linear programming model by using a linearization technique with superior expressions. Since the reformulation has a large number of decision variables and constraints, we integrate SAA with a dual decomposition Lagrangian relaxation technique (DDLR) to develop a solution method called SAA-DDLR. Numerical experiments are conducted to illustrate the effectiveness and applicability of our model and method. Sensitivity analysis reveals that both the 4PL and 3PLs can benefit from the quantity discount scheme. Managerial insights are drawn for the 4PL to run a cost-effective logistics system in the presence of quantity discounts. [ABSTRACT FROM AUTHOR]

Due to the increasingly competitive nature of the global market, the capability of controlling delivery time is becoming a significant advantage for enterprises. A novel fourth-party logistics (4PL) network design problem with the objective of minimizing the overall cost under service time constraint and stochastic demand is proposed in the paper. To address this problem, a two-stage nonlinear stochastic programming model is proposed. The topological structure of the 4PL network is decided in the first stage, while the network flows are determined in the second stage. By using auxiliary variables to linearize the service time constraint and by adopting the sample average approximation (SAA) method to handle the stochastic demand, the two-stage nonlinear stochastic programming model is transformed into a mixed integer linear programming (MILP) model. To overcome the difficulties of solving the MILP model caused by a large number of demand scenarios and integer-valued decision variables, a variable separation (VS) strategy is presented to improve the dual decomposition and Lagrangian relaxation (DDLR) approach to propose a VSDDLR-SAA algorithm. Results of the numerical examples and a real-life case illustrate the effectiveness of the proposed model and VSDDLR-SAA algorithm. Comparison analysis of the 4PL network and the supply chain network shows that 4PL can deliver products within the prescribed time at a lower cost by cooperating with third-party logistics providers. [ABSTRACT FROM AUTHOR]