Narrow Results

Redundancy in constraints and variables are usually studied in linear, integer and non-linear programming problems. However, main emphasis has so far been given only to linear programming problems. In this paper, an algorithm that identifies redundant objective functions in multi-objective stochastic fractional programming problems is provided. A solution procedure is also illustrated. This reduces the number of objective functions in cases where redundant objective functions exist.

Deterministic models, even if used repeatedly, will not capture the essence of planning in an uncertain world. Flexibility and robustness can only be properly valued in models that use stochastics explicitly, such as stochastic optimization models. However, it may also be very important to capture how the random phenomena are related to one another. In this article we show how the solution to a stochastic service network design model depends heavily on the correlation structure among the random demands. The major goal of this paper is to discuss why this happens, and to provide insights into the effects of correlations on solution structures. We illustrate by an example.

We consider a chance-constrained quadratic knapsack problem (CQKP) where each item has a random size that is finitely distributed. We present a new convex 0-1 quadratic program reformulation for CQKP. This new reformulation improves the existing reformulation for general 0-1 quadratic program based on diagonal perturbation in the sense that the continuous relaxation of the new reformulation is tighter than or at least as tight as that of the existing reformulation. The improved reformulation is derived from a general matrix decomposition of the objective function and piecewise linearization of 0-1 variables. We show that the optimal parameters in the improved reformulation can be obtained by solving an SDP problem. Extension to k-item probabilistic quadratic knapsack problems is also discussed. Preliminary comparison results are reported to demonstrate the effectiveness of the improved reformulation.

We focus on rating of non-life insurance contracts. We employ multiplicative models with basic premium levels and specific surcharge coefficients for various levels of selected risk/rating factors. We use generalized linear models (GLM) to describe the probability distribution of total losses for a contract during one year. We show that the traditional frequency–severity approaches based only on GLM with logarithmic link function can lead to estimates which do not fulfill business requirements. For example, a maximal surcharge and monotonicity of coefficient can be desirable. Moreover, our approach can handle total losses, which are based on arbitrary loss distributions, possibly decomposed into several classes, e.g., small and large or property and bodily injury. Various costs and loadings can be also incorporated into the tariff rates. We propose optimization problems for rate estimation which enable hedging against expected losses and taking into account a prescribed loss ratio and other business requirements. Moreover, we introduce stochastic programming problems with reliability type constraints which take into account individual risk of each rate cell or collective risk. In the numerical study, we apply the approaches to Motor Third Party Liability (MTPL) policies.

Although many researches have conjectured the optimality of the hedging point policy for the multiple-product stochastic capacitated periodic review problem, it is still a challenge to prove the hypothesis. This paper considers a special case of the capacitated multiple-product periodic review problem where stochastic demand distribution, production rate, unit production cost and periodic expected inventory cost are the same for all different products. For this symmetric problem, we prove an optimal policy where ordering and non-ordering regions for every product are defined. Our research provides significant implications to the characterization of the optimal policy for the general problem.

In this paper, we consider the quantitative stability analysis and empirical approximation of risk-averse models induced by two-stage stochastic programs with full random recourse. We first establish the quantitative stability under the mean-coherent risk measure framework and the expected utility framework, respectively, under suitable probability metrics. Based on the obtained quantitative stability results, we then investigate the empirical approximation to these models, and estimate the rates of convergence for the optimal value and optimal solution set with the aid of Ky Fan distance.

In this paper, we study the problem of targeting a set of individuals to trigger a cascade of influence in a social network such that the influence diffuses to all individuals with the minimum time, given that the cost of initially influencing each individual is with randomness and that the budget available is limited. We adopt the incremental chance model to characterize the diffusion of influence, and propose three stochastic programming models that correspond to three different decision criteria respectively. A modified greedy algorithm is presented to solve the proposed models, which can flexibly trade off between solution performance and computational complexity. Experiments are performed on random graphs, by which we show that the algorithm we present is effective.

We present a stochastic optimization model for hotel revenue management with multiple-day stays under an uncertain environment. Since a decision maker may face several scenarios when renting out rooms, we use a semi-absolute deviation model to measure the risk of hotel revenue, and only consider the risk of falling below the expected revenue. The method proposed in this paper can be changed to a linear programming model by applying linearization techniques. Some examples are presented to illustrate the efficiency of this method.

Nowadays, due to environmental issues, government rules and economic interests have increased attention to the collection and recovery of products, which has led to the formation of new concepts such as reverse and closed-loop supply chains. The implementation of the closed-loop supply chain as a solution to sustainable development is expanding from one hand and increasing the profitability of companies on the other. For this purpose, a mathematical model was developed to design an integrated closed-loop supply chain network, which is a combination of two-problem localization problems and flow optimization. The proposed model was designed to minimize network costs and to maximize the level of responsiveness to customers. The cost parameters of establishing centers in this model are uncertain; to overcome the model’s uncertainties, stochastic programming is used. In the mathematical model, supplier, manufacturer, distributor and customer in the direct supply chain and collection/rehabilitation, destruction, recycling centers and, second-type distribution center for sale of second-hand products as well as second-hand products customers in the reverse flow are considered, to be closer to the real today world. This model is multi-periodic mix integer nonlinear programming where the shortage has allowed. To motivate and encourage customers to buy more, in addition to getting closer to the real world and it happens more in practice, is considered all units of discount for transportation cost in the forward flow. To solve this model Non-Dominated Sorting Genetic Algorithm II (NSGA-II) and Multi-Objective Particle Swarm Optimization (MOPSO) is using. The parameter tuning was done using the Taguchi method. Then, the important criteria for measurement and comparison of performance algorithms have used, including the Mean Ideal Distance, Diversification Metric, Number of Pareto-optimal Solutions, and the Quality Metric. Results of the Comparative metrics show that NSGA-II outperforms MOPSO in almost all cases in achieving the best trade-off solutions.

How does risk and uncertainty in climate thresholds impact optimal short-run mitigation? This paper contrasts the near-term mitigation consequences of using an expected value, stochastic programming, and stochastic control model to capture the policy effects of uncertain climate thresholds. The risk of threshold outcomes increases expected climate damages. The passive learning associated with stochastic programming creates an extra incentive to mitigate promptly by reducing the damages from remaining threshold hazards. The active learning associated with stochastic control creates yet another incentive to do near-term mitigation, through the delaying of potential threshold effects.

In power delivery systems, the use of dispersed generation and security control to improve network utilization requires the optimal use of system control devices. The installation of loop controller allows the distribution system to operate in a loop configuration, achieving effective management of voltage and power flow. In the investment planning process, it is important to identify the optimal location and installed capacity of the equipment such that all operational constraints are satisfied. This chapter presents a method for identifying the optimal location and capacity with the minimum installation cost. Our novel approach uses an economic model that considers the fixed costs. A slope scaling procedure is presented, and its efficiency is demonstrated using numerical experiments.

For a long time modeling approaches to stochastic programming were dominated by scenario generation methods. Consequently the main computational effort went into development of decomposition type algorithms for solving constructed large scale (linear) optimization problems. A different point of view emerged recently where computational complexity of stochastic programming problems was investigated from the point of view of randomization methods based on Monte Carlo sampling techniques. In that approach the number of scenarios is irrelevant and can be infinite. On the other hand, from that point of view there is a principle difference between computational complexity of two and multistage stochastic programming problems – certain classes of two stage stochastic programming problems can be solved with a reasonable accuracy and reasonable computational effort, while (even linear) multistage stochastic programming problems seem to be computationally intractable in general.

In this paper sensitivity analysis is adopted in order to reveal the role of randomness of a stochastic second-order cone program (Maggioni et al., 2009) for mobile ad-hoc networks starting from the semidefinite stochastic locationaided routing (SLAR) model, described in Ariyawansa and Zhu (2006) and Zhu et al. (2011). The algorithm looks for a destination node and sets up a route by means of the expected zone, the region where the sender node expects to find the destination node and the requested zone defined by the sender node for spreading the route request to the destination node. The movements of the destination node are represented by ellipses scenarios, randomly generated by uniform and normal distributions in a neighborhood of the initial position of the destination node. Sensitivity analysis is performed by considering an increasing number of scenarios, different costs of flooding and latency penalty. Evaluation of Expected Value of Perfect Information EVPI and Value of Stochastic Solution VSS (Maggioni and Wallace, 2010; Birge, 1970) allows us to find the range of values in which it is convenient the deterministic versus the stochastic approach.