Capital rationing problems under uncertainty and risk

Capital rationing is a major problem in managerial decision making. The classical mathematical formulation of the problem relies on a multi-dimensional knapsack model with known input parameters. Since capital rationing is carried out in conditions where uncertainty is the rule rather than the exception, the hypothesis of deterministic data limits the applicability of deterministic formulations in real settings. This paper proposes a stochastic version of the capital rationing problem which explicitly accounts for uncertainty. In particular, a mathematical formulation is provided in the framework of stochastic programming with joint probabilistic constraints and a novel solution approach is proposed. The basic model is also extended to include specific risk measures. Preliminary computational results are presented and discussed.     

The mixed capacitated general routing problem under uncertainty

We study the General Routing Problem defined on a mixed graph and with stochastic demands. The problem under investigation is aimed at finding the minimum cost set of routes to satisfy a set of clients whose demand is not deterministically known. Since each vehicle has a limited capacity, the demand uncertainty occurring at some clients affects the satisfaction of the capacity constraints, that, hence, become stochastic. The contribution of this paper is twofold: firstly we present a chance-constrained integer programming formulation of the problem for which a deterministic equivalent is derived. The introduction of uncertainty into the problem poses severe computational challenges addressed by the design of a branch-and-cut algorithm, for the exact solution of limited size instances, and of a heuristic solution approach exploring promising parts of the search space. The effectiveness of the solution approaches is shown on a probabilistically constrained version of the benchmark instances proposed in the literature for the mixed capacitated general routing problem.     

Robust empty container repositioning considering foldable containers

Because of the extreme imbalance in intercontinental trade, the repositioning of empty containers creates a significant problem for shipping companies. There are many efforts to reduce the cost of repositioning empty containers, one of which is a foldable container. This paper proposes a robust formulation for the empty container repositioning problem considering foldable containers under demand uncertainty. The robust formulation can be used as a tractable approximation of a multistage stochastic programming formulation which is computationally intractable. Moreover, the robust formulation requires only limited information about the distribution of demand to replicate real-world situations. Computational results show that the proposed formulation performs well in terms of operating costs and there exists a significant cost-saving effect when foldable containers are used in maritime transportation.     

Knapsack problem with probability constraints

This paper is dedicated to a study of different extensions of the classical knapsack problem to the case when different elements of the problem formulation are subject to a degree of uncertainty described by random variables. This brings the knapsack problem into the realm of stochastic programming. Two different model formulations are proposed, based on the introduction of probability constraints. The first one is a static quadratic knapsack with a probability constraint on the capacity of the knapsack. The second one is a two-stage quadratic knapsack model, with recourse, where we introduce a probability constraint on the capacity of the knapsack in the second stage. As far as we know, this is the first time such a constraint has been used in a two-stage model. The solution techniques are based on the semidefinite relaxations. This allows for solving large instances, for which exact methods cannot be used. Numerical experiments on a set of randomly generated instances are discussed below.     

A Mixed Integer Programming Formulation for the Total Flow Time Single Machine Robust Scheduling Problem with Interval Data

We consider a version of the total flow time single machine scheduling problem where uncertainty about processing times is taken into account. Namely an interval of equally possible processing times is considered for each job, and optimization is carried out according to a robustness criterion. We propose the first mixed integer linear programming formulation for the resulting optimization problem and we explain how some known preprocessing rules can be translated into valid inequalities for this formulation. Computational results are finally presented. Work funded by the Swiss National Science Foundation through project 200020-109854/1.     

Equality Constraints in Multiobjective Robust Design Optimization: Decision Making Problem

Robust design optimization (RDO) problems can generally be formulated by incorporating uncertainty into the corresponding deterministic problems. In this context, a careful formulation of deterministic equality constraints into the robust domain is necessary to avoid infeasible designs under uncertain conditions. The challenge of formulating equality constraints is compounded in multiobjective RDO problems. Modeling the tradeoffs between the mean of the performance and the variation of the performance for each design objective in a multiobjective RDO problem is itself a complex task. A judicious formulation of equality constraints adds to this complexity because additional tradeoffs are introduced between constraint satisfaction under uncertainty and multiobjective performance. Equality constraints under uncertainty in multiobjective problems can therefore pose a complicated decision making problem. In this paper, we provide a new problem formulation that can be used as an effective multiobjective decision making tool, with emphasis on equality constraints. We present two numerical examples to illustrate our theoretical developments.     

Capacity allocation problem with random demands for the rail container carrier

In this paper, we consider the formulation and heuristic algorithm for the capacity allocation problem with random demands in the rail container transportation. The problem is formulated as the stochastic integer programming model taking into account matches in supply and demand of rail container transportation. A heuristic algorithm for the stochastic integer programming model is proposed. The solution to the model is found by maximizing the expected total profit over the possible control decisions under the uncertainty of demands. Finally, we give numerical experiments to demonstrate the efficiency of the heuristic algorithm.     

A two-stage stochastic programming framework for transportation planning in disaster response

This study proposes a two-stage stochastic programming model to plan the transportation of vital first-aid commodities to disaster-affected areas during emergency response. A multi-commodity, multi-modal network flow formulation is developed to describe the flow of material over an urban transportation network. Since it is difficult to predict the timing and magnitude of any disaster and its impact on the urban system, resource mobilization is treated in a random manner, and the resource requirements are represented as random variables. Furthermore, uncertainty arising from the vulnerability of the transportation system leads to random arc capacities and supply amounts. Randomness is represented by a finite sample of scenarios for capacity, supply and demand triplet. The two stages are defined with respect to information asymmetry, which discloses uncertainty during the progress of the response. The approach is validated by quantifying the expected value of perfect and stochastic information in problem instances generated out of actual data.     

Robust multiobjective optimization & applications in portfolio optimization

Motivated by Markowitz portfolio optimization problems under uncertainty in the problem data, we consider general convex parametric multiobjective optimization problems under data uncertainty. For the first time, this uncertainty is treated by a robust multiobjective formulation in the gist of Ben-Tal and Nemirovski. For this novel formulation, we investigate its relationship to the original multiobjective formulation as well as to its scalarizations. Further, we provide a characterization of the location of the robust Pareto frontier with respect to the corresponding original Pareto frontier and show that standard techniques from multiobjective optimization can be employed to characterize this robust efficient frontier. We illustrate our results based on a standard mean–variance problem.     

Reliability-based user equilibrium in a transport network under the effects of speed limits and supply uncertainty

This study investigates the system-wide traffic flow re-allocation effect of speed limits in uncertain environments. Previous studies have only considered link capacity degradation, which is only one of the factors that lead to supply uncertainty. This study examines how imposing speed limits reallocates the traffic flows in a situation of general supply uncertainty with risk-averse travelers. The effects of imposing a link-specific speed limit on link driving speed and travel time are analyzed, given the link travel time distribution before imposing the speed limit. The expected travel time and travel time standard deviation of a link with a speed limit are derived from the link travel time distribution and are both continuous, monotone, and convex functions in terms of link flow. A distribution-free, reliability-based user equilibrium with speed limits is established, in which travelers are assumed to choose routes that minimize their own travel time budget. A variational inequality formulation for the equilibrium problem is proposed and the solution properties are provided. In this study, the inefficiency of a reliability-based user equilibrium flow pattern with speed limits is defined and found to be bounded above when supply uncertainty refers to capacity degradation. The upper bound depends on the level of risk aversion of travelers, a ratio related to the design and worst-case link capacities, and the highest power of all link performance functions.     

Robust solutions for network design under transportation cost and demand uncertainty

In many applications, the network design problem (NDP) faces significant uncertainty in transportation costs and demand, as it can be difficult to estimate current (and future values) of these quantities. In this paper, we present a robust optimization-based formulation for the NDP under transportation cost and demand uncertainty. We show that solving an approximation to this robust formulation of the NDP can be done efficiently for a network with single origin and destination per commodity and general uncertainty in transportation costs and demand that are independent of each other. For a network with path constraints, we propose an efficient column generation procedure to solve the linear programming relaxation. We also present computational results that show that the approximate robust solution found provides significant savings in the worst case while incurring only minor sub-optimality for specific instances of the uncertainty.     

A simple plant-location model for quantity-setting firms subject to price uncertainty

The uncapacitated plant location problem under uncertainty is formulated in a mean-variance framework with prices in various markets correlated via their response to a common random factor. This formulation results in a mixed-integer quadratic programming problem. However, for a given integer solution, the resulting quadratic programming problem is amenable to a very simple solution procedure. The simplicity of this algorithm means that reasonably large problems should be solvable using existing branch-and-bound techniques.     

A Robust Optimization Approach to Dynamic Pricing and Inventory Control with no Backorders

In this paper, we present a robust optimization formulation for dealing with demand uncertainty in a dynamic pricing and inventory control problem for a make-to-stock manufacturing system. We consider a multi-product capacitated, dynamic setting. We introduce a demand-based fluid model where the demand is a linear function of the price, the inventory cost is linear, the production cost is an increasing strictly convex function of the production rate and all coefficients are time-dependent. A key part of the model is that no backorders are allowed. We show that the robust formulation is of the same order of complexity as the nominal problem and demonstrate how to adapt the nominal (deterministic) solution algorithm to the robust problem.     

Models for single-sector stochastic air traffic flow management under reduced airspace capacity

Air traffic efficiency is heavily influenced by unanticipated factors that result in capacity reduction. Of these factors, weather is the most significant cause of delays in airport and airspace operations. Considering weather-related uncertainty, air traffic flow management involves controlling air traffic through allocation of available airspace capacity to flights. The corresponding decision process results in a stochastic dynamic problem where aircraft on the ground and in the air are controlled based on the evolution of weather uncertainty. We focus on the single-sector version of the problem that is applicable to a majority of cases where a volume of airspace has reduced capacity due to convective weather. We model the decision process through stochastic integer programming formulations and computationally analyse it for tractability. We then demonstrate through actual flight schedule data that a simplistic but practically implementable approximation procedure is a generally effective solution approach for these models.     

Stochastic Nonlinear Complementarity Problem and?Applications to?Traffic Equilibrium under?Uncertainty

The expected residual minimization (ERM) formulation for the stochastic nonlinear complementarity problem (SNCP) is studied in this paper. We show that the involved function is a stochastic R ₀ function if and only if the objective function in the ERM formulation is coercive under a mild assumption. Moreover, we model the traffic equilibrium problem (TEP) under uncertainty as SNCP and show that the objective function in the ERM formulation is a stochastic R ₀ function. Numerical experiments show that the ERM-SNCP model for TEP under uncertainty has various desirable properties. This work was partially supported by a Grant-in-Aid from the Japan Society for the Promotion of Science. The authors thank Professor Guihua Lin for pointing out an error in Proposition 2.1 on an earlier version of this paper. The authors are also grateful to the referees for their insightful comments.     

Extension of some results for channel capacity using a generalized information measure

A new formulation for the channel capacity problem is derived by using the duality theory of convex programming. The simple nature of this dual representation is suitable for computational purposes. The results are derived in a unified way by formulating the channel capacity problem as a special case of a general class of concave programming problems involving a generalized information measure recently introduced by Burbea and Rao [10].Research supported by National Science Foundation Grant No. ECS-8604354.     

Decomposition for adjustable robust linear optimization subject to uncertainty polytope

We present in this paper a general decomposition framework to solve exactly adjustable robust linear optimization problems subject to polytope uncertainty. Our approach is based on replacing the polytope by the set of its extreme points and generating the extreme points on the fly within row generation or column-and-row generation algorithms. The novelty of our approach lies in formulating the separation problem as a feasibility problem instead of a max–min problem as done in recent works. Applying the Farkas lemma, we can reformulate the separation problem as a bilinear program, which is then linearized to obtained a mixed-integer linear programming formulation. We compare the two algorithms on a robust telecommunications network design under demand uncertainty and budgeted uncertainty polytope. Our results show that the relative performance of the algorithms depend on whether the budget is integer or fractional.     

Timing order fulfillment of capital goods under a constrained capacity

This paper studies the order-fulfillment process of a supplier producing multiple customized capital goods. The times when orders are confirmed by customers are random. The supplier can only work on one product at any time due to capacity constraints. The supplier must determine the optimal time to start the process for each order so that the total expected cost of having the goods ready before or after their orders are confirmed is minimized. We formulate this problem as a discrete time Markov decision process. The optimal policy is complex in general. It has a threshold-type structure and can be fully characterized only for some special cases. Based on our formulation, we compute the optimal policy and quantify the value of jointly managing the order fulfillment processes of multiple orders and the value of taking into account demand arrival time uncertainty.     

Computations with disjunctive cuts for two-stage stochastic mixed 0-1 integer programs

Two-stage stochastic mixed-integer programming (SMIP) problems with recourse are generally difficult to solve. This paper presents a first computational study of a disjunctive cutting plane method for stochastic mixed 0-1 programs that uses lift-and-project cuts based on the extensive form of the two-stage SMIP problem. An extension of the method based on where the data uncertainty appears in the problem is made, and it is shown how a valid inequality derived for one scenario can be made valid for other scenarios, potentially reducing solution time. Computational results amply demonstrate the effectiveness of disjunctive cuts in solving several large-scale problem instances from the literature. The results are compared to the computational results of disjunctive cuts based on the subproblem space of the formulation and it is shown that the two methods are equivalently effective on the test instances.     

Radar placement based on a geometric uncertainty multiplier reduction criterion

In this paper the problem of retrieving wind field information from Doppler radar data motivates the formulation of a method to design radar network configurations. The problem of estimating wind velocities from radar data is posed and used to construct a certain retrieval operator. This operator contains a factor that may be interpreted as an uncertainty multiplier. It depends on the geometry of the configuration of the radar network. The uncertainty multiplier is shown to vary continuously with perturbations of the network configuration. It is also shown to be a generalization of the Doppler angle condition used in meteorology. Numerical examples are presented to determine a network of five radars minimizing the uncertainty multiplier for the problem. Also, a configuration of sites is determined that maximizes the area of the Doppler region.