Schumann,a modeling framework for supply chain management under uncertainty

We present a modeling framework for the optimization of a manufacturing, assembly and distribution (MAD) supply chain planning problem under uncertainty in product demand and component supplying cost and delivery time, mainly. The automotive sector has been chosen as the pilot area for this type of multiperiod multiproduct multilevel problem, but the approach has a far more reaching application. A deterministic treatment of the problem provides unsatisfactory results. We use a 2-stage scenario analysis based on a partial recourse approach, where MAD supply chain policy can be implemented for a given set of initial time periods, such that the solution for the other periods does not need to be anticipated and, then, it depends on the scenario to occur. In any case, it takes into consideration all the given scenarios. Very useful schemes are used for modeling balance equations and multiperiod linking constraints. A dual approach splitting variable scheme is been used for dealing with the implementable time periods related variables, via a redundant circular linking representation.     

WARSYP: a robust modeling approach for water resources system planning under uncertainty

In this work we present a solution procedure for multiperiod water resources system planning, where the aim is to obtain the optimal policy for water resources utilization under uncertainty. The target levels to be achieved are related to the following parameters: reservoir capacity, hydropower demand and other demand uses for urban, industrial, irrigation, ecological and other purposes. The approach allows for the conjunctive use of surface systems together with groundwater. The hydrological exogenous inflow and demand of different kinds are considered via a scenario analysis scheme due to the uncertainty of the parameters. So, a multistage scenario tree is generated and, through the use of full recourse techniques, an implementable solution is obtained for each scenario group at each stage along the planning horizon. A novel scheme is presented for modeling the constraints to preserve the reserved stored water in (directly and non-directly) upstream reservoirs to satisfy potential future needs in demand centers at given time periods. An algorithmic framework based on augmented Lagrangian decomposition is presented. Computational experience is reported for the deterministic case.     

International portfolio management with affine policies

While dynamic decision making has traditionally been represented as scenario trees, these may become severely intractable and difficult to compute with an increasing number of time periods. We present an alternative tractable approach to multiperiod international portfolio optimization based on an affine dependence between the decision variables and the past returns. Because local asset and currency returns are modeled separately, the original model is non-linear and non-convex. With the aid of robust optimization techniques, however, we develop a tractable semidefinite programming formulation of our model, where the uncertain returns are contained in an ellipsoidal uncertainty set. We add to our formulation the minimization of the worst case value-at-risk and show the close relationship with robust optimization. Numerical results demonstrate the potential gains from considering a dynamic multiperiod setting relative to a single stage approach.     

Financial scenario generation for stochastic multi-stage decision processes as facility location problems

The quality of multi-stage stochastic optimization models as they appear in asset liability management, energy planning, transportation, supply chain management, and other applications depends heavily on the quality of the underlying scenario model, describing the uncertain processes influencing the profit/cost function, such as asset prices and liabilities, the energy demand process, demand for transportation, and the like. A common approach to generate scenarios is based on estimating an unknown distribution and matching its moments with moments of a discrete scenario model. This paper demonstrates that the problem of finding valuable scenario approximations can be viewed as the problem of optimally approximating a given distribution with some distance function. We show that for Lipschitz continuous cost/profit functions it is best to employ the Wasserstein distance. The resulting optimization problem can be viewed as a multi-dimensional facility location problem, for which at least good heuristic algorithms exist. For multi-stage problems, a scenario tree is constructed as a nested facility location problem. Numerical convergence results for financial mean-risk portfolio selection conclude the paper.     

A Note on the Application of Dynamic Correction to the Optimization of a Multiperiod Resource Allocation Problem

In this note, Luss's approach to optimization of a multiperiod resource allocation problem is employed to include carry-over effects of the controllable variable on future periods and also to include the carry-over effect from previous periods.     

Knowledge‐based scenario tree generation methods and application in multiperiod portfolio selection problem

A scenario tree is an efficient way to represent a stochastic data process in decision problems under uncertainty. This paper addresses how to efficiently generate appropriate scenario trees. A knowledge‐based scenario tree generation method is proposed; the new method is further improved by accounting for subjective judgements or expectations about the random future. Compared with existing approaches, complicated mathematical models and time‐consuming estimation, simulation and optimization problem solution are avoided in our knowledge‐based algorithms, and large‐scale scenario trees can be quickly generated. To show the advantages of the new algorithms, a multiperiod portfolio selection problem is considered, and a dynamic risk measure is adopted to control the intermediate risk, which is superior to the single‐period risk measure used in the existing literature. A series of numerical experiments are carried out by using real trading data from the Shanghai stock market. The results show that the scenarios generated by our algorithms can properly represent the underlying distribution; our algorithms have high performance, say, a scenario tree with up to 10,000 scenarios can be generated in less than a half minute. The applications in the multiperiod portfolio management problem demonstrate that our scenario tree generation methods are stable, and the optimal trading strategies obtained with the generated scenario tree are reasonable, efficient and robust. Copyright © 2013 John Wiley & Sons, Ltd.     

On distributionally robust multiperiod stochastic optimization

This paper considers model uncertainty for multistage stochastic programs. The data and information structure of the baseline model is a tree, on which the decision problem is defined. We consider “ambiguity neighborhoods” around this tree as alternative models which are close to the baseline model. Closeness is defined in terms of a distance for probability trees, called the nested distance. This distance is appropriate for scenario models of multistage stochastic optimization problems as was demonstrated in Pflug and Pichler (SIAM J Optim 22:1–23, 2012). The ambiguity model is formulated as a minimax problem, where the the optimal decision is to be found, which minimizes the maximal objective function within the ambiguity set. We give a setup for studying saddle point properties of the minimax problem. Moreover, we present solution algorithms for finding the minimax decisions at least asymptotically. As an example, we consider a multiperiod stochastic production/inventory control problem with weekly ordering. The stochastic scenario process is given by the random demands for two products. We determine the minimax solution and identify the worst trees within the ambiguity set. It turns out that the probability weights of the worst case trees are concentrated on few very bad scenarios.     

Joint pricing and ordering policy for an advance booking system with partial order cancellations

In this paper, an inventory model for deteriorating items with price-dependent demand is developed. The cycle is divided into two periods, where an advance sales period is followed by a spot sales period. In practice, customers with reservations may cancel their orders before receiving them. During the advance sales period, the rate of reservations which will not be cancelled is dependent on the length of the waiting time for the receiving order. During the spot sales period, all customers receive their orders at the time of the purchase. We prove the existence of the realistic relationship that the advance sales price is smaller than the spot sales price. We also develop some useful properties and provide an iterative procedure for solving the maximization problem. Numerical examples are given to demonstrate the effectiveness of the proposed approach and we conclude the paper with suggestions for possible future research.     

Study of cross-diffusion induced Turing patterns in a ratio-dependent prey-predator model via amplitude equations

Cross-diffusion models the situation where the presence, absence or abundance of one species of population affects the movement of other species of population in the domain under consideration and vice versa. Inclusion of cross-diffusion terms makes the modeling approach more realistic and shows significant impact on the spatio-temporal pattern formation scenario. In this paper, cross-diffusion is considered in a prey-predator model with ratio-dependent functional response, in addition to self-diffusion. Weakly nonlinear analysis is used near the Turing bifurcation boundary to derive the amplitude equations. From the stability analysis of the amplitude equations, conditions for emergence of Turing patterns such as cold spot, hot spot, mixture of spots and stripes and labyrinthine are identified. The analytical results are then verified with the help of numerical simulations. Results are general in nature and can be used to study the effect of cross-diffusion on other prey predator models both analytically and numerically.     

An N-path user equilibrium for transportation networks

The traditional trip-based approach to transportation modeling has been employed for the past decade. The last step of the trip-based modeling approach is traffic assignment, which has been typically formulated as a user equilibrium (UE) problem. In the conventional perspective, the definition of UE traffic assignment is the condition that no road user can unilaterally change routes to reduce their travel time. An equivalent definition is that the travel times of all the used paths between any given origin–destination pair are equal and less than those of the unused paths. The underlying assumption of the UE definition is that road users have full information on the available transportation paths and can potentially use any path if the currently used path is overly congested. However, a more practical scenario is that each road user has a limited path set within which she/he can choose routes from. In this new scenario, we call the resulting user equilibrium an N-path user equilibrium (NPUE), in which each road user has only N paths to select from when making route choices in the network. We introduce a new formulation of the NPUE and derive optimality conditions based on this formulation. Different from traditional modeling framework, the constraints of the proposed model are of linear form, which makes it possible to solve the problem with conventional convex programming techniques. We also show that the traditional UE is a special case of an NPUE and prove the uniqueness of the resulting flow pattern of the NPUE. To efficiently solve this problem, we devise path-based and link-based solution algorithms. The proposed solution algorithms are empirically applied to networks of various sizes to examine the impact of constrained user path sets. Numerical results demonstrate that NPUE results can differ significantly from UE results depending on the number of paths available to road users. In addition, we observed an interesting phenomenon, where increasing the number of paths available to road users can sometimes decrease the overall system performance due to their selfish routing behaviors. This paradox demonstrates that network information should be provided with caution, as such information can do more harm than good in certain transportation systems.     

Dynamic Network Flow with Uncertain Arc Capacities: Decomposition Algorithm and Computational Results

In a multiperiod dynamic network flow problem, we model uncertain arc capacities using scenario aggregation. This model is so large that it may be difficult to obtain optimal integer or even continuous solutions. We develop a Lagrangian decomposition method based on the structure recently introduced in G.D. Glockner and G.L. Nemhauser, Operations Research, vol. 48, pp. 233–242, 2000. Our algorithm produces a near-optimal primal integral solution and an optimum solution to the Lagrangian dual. The dual is initialized using marginal values from a primal heuristic. Then, primal and dual solutions are improved in alternation. The algorithm greatly reduces computation time and memory use for real-world instances derived from an air traffic control model.     

线性可替代资源的多阶段分配模型与求解

可替代资源是指在产品生产过程中具有相同功能且能相互替代的资源。本文根据可替代资源之间通常具有的级性替代关系，建立了线性可替代资源的多阶段分配模型，并在产品生产水平与产品需求的加权相对偏最小的目标下，给出了求模型优解的方法．     

Strategic financial risk management and operations research

Risk management has become a vital topic for financial institutions in the 1990s. Strategically, asset/liability management systems are important tools for controlling a firm's financial risks. They manage these risks by dynamically balancing the firm's asset and liabilities to achieve the firm's objectives. We discuss such leading international firms as Towers Perrin, Frank Russell, and Falcon Asset Management, which apply asset/liability management for efficiently managing risk over extended time periods. Three components of asset/liability management are described: 1) a multi-stage stochastic program for coordinating the asset/liability decisions; 2) a scenario generation procedure for modeling the stochastic parameters; and 3) solution algorithms for solving the resulting large-scale optimization problem.     

Exact and hybrid methods for the multiperiod field service routing problem

This article deals with a particular class of routing problem, consisting of the planning and routing of technicians in the field. This problem has been identified as a multiperiod, multidepot uncapacitated vehicle routing problem with specific constraints that we call the multiperiod field service routing problem (MPFSRP). We propose a set covering formulation of the problem for the column generation technique and we develop an exact branch and price solution method for small-sized instances. We also propose several heuristic versions for larger instances. We present the results of experiments on realistic data adapted from an industrial application.     

Weak aggregating algorithm for the distribution-free perishable inventory problem

We formulate the multiperiod, distribution-free perishable inventory problem as a problem of prediction with expert advice and apply an online learning method (the Weak Aggregating Algorithm) to solve it. We show that the asymptotic average performance of this method is as good as that of any time-dependent stocking rule in a given parametric class.     

Capacitated,balanced, sequential location-allocation problems on chains and trees

This paper considers finite horizon, multiperiod, sequential, minisum location-allocation problems on chain graphs and tree networks. The demand has both deterministic and probabilistic components, and increases dynamically from period to period. The problem is to locate one additionalcapacitated facility in each of thep specified periods, and to determine the service allocations of the facilities, in order to optimally satisfy the demand on the network. In this context, two types of objective criteria or location strategies are addressed. The first is a myopic strategy in which the present period cost is minimized sequentially for each period, and the second is a discounted present worth strategy. For the chain graph, we analyze ap-facility problem under both these criteria, while for the tree graph, we analyze a 3-facility myopic problem, and a 2-facility discounted present worth problem. All these problems are nonconvex, and we specify a finite set of candidate solutions which may be compared in order to determine a global optimal solution.     

Stochastic network optimization models for investment planning

We describe and compare stochastic network optimization models for investment planning under uncertainty. Emphasis is placed on multiperiod a sset allocation and active portfolio management problems. Myopic as well as multiple period models are considered. In the case of multiperiod models, the uncertainty in asset returns filters into the constraint coefficient matrix, yielding a multi-scenario program formulation. Different scenario generation procedures are examined. The use of utility functions to reflect risk bearing attitudes results in nonlinear stochastic network models. We adopt a newly proposed decomposition procedure for solving these multiperiod stochastic programs. The performance of the models in simulations based on historical data is discussed.Research partially supported by National Science Foundation Grant No. DCR-861-4057 and IBM Grant No. 5785. Also, support from Pacific Financial Companies is gratefully acknowledged.     

Network simplex algorithm for the general equal flow problem

In this paper the general equal flow problem is considered. This is a minimum cost network flow problem with additional side constraints requiring the flow of arcs in some given sets of arcs to take on the same value. This model can be applied to approach water resource system management problems or multiperiod logistic problems in general involving policy restrictions which require some arcs to carry the same amount of flow through the given study period. Although the bases of the general equal flow problem are no longer spanning trees, it is possible to recognize a similar structure that allows us to take advantage of the practical computational capabilities of network models. After characterizing the bases of the problem as good (r+1)-forests, a simplex primal algorithm is developed that exploits the network structure of the problem and requires only slight modifications of the well-known network simplex algorithm.     

Probabilistic modeling of multiperiod service levels

This study investigates multiperiod service level (MSL) policies in supply chains facing a stochastic customer demand. The objective of the supply chains is to construct integrated replenishment plans that satisfy strict stockout-oriented performance measures which apply across a multiperiod planning horizon. We formulate the stochastic service level constraints for the fill rate, ready rate, and conditional expected stockout MSL policies. The modeling approach is based on the concept of service level trajectory and provides reformulations of the stochastic planning problems associated with each MSL policy that can be efficiently solved with off-the-shelf optimization solvers. The approach enables the handling of correlated and non-stationary random variables, and is flexible enough to accommodate the implementation of fair service level policies, the assignment of differentiated priority levels per products, or the introduction of response time requirements. We use an earthquake disaster management case study to show the applicability of the approach and derive practical implications about service level policies.     

Optimization of a Multiperiod Resource Allocation Model

In this paper we examine multiperiod resource allocation problems, such as allocating a given marketing budget among T periods. The return functions of each period are assumed to be concave functions of the effective effort variable, which is composed of the expenditures in all previous periods and the present one. Assuming that the effect of an amount spent in period t is decreasing by a fixed rate in successive periods, necessary and sufficient conditions for a non-boundary optimal policy are derived. Under these conditions the optimal policy which maximizes total returns is obtained.