The distribution of the product of two triangular random variables

Although computer simulation can be used to determine the distribution of the product of two triangularly distributed variables for a specific application, a closed-form expression is preferable in general. Assuming independence, the closed-form probability density function of this product is derived.

New directions for higher education

Publ. comme no 137, spring 2007 de la revue New directions for higher educationIndexBibliogr. à la fin des texte

Risks of Catastrophic Derailments Involving the Release of Hazardous Materials

Models are formulated and computed to assess the risks of hazardous materials releases in train derailments in terms of: the probability of any number of fatalities in an accident, the probability of any total number of fatalities from all the accidents in a year, and the frequency of accidents which result in any given number of fatalities. These functions are evaluated using data bases and analytical methods which provide estimates of exposure levels (traffic volumes, track conditions and population densities), spill occurrence and spill size probabilities, and fatal spill impacts (the size of the lethal area in any given accident scenario). Despite the sparsity of historical data, this methodology enables us to express the potential for catastrophic occurrences in quantitative terms, to compare these numbers with estimates of other risks, and to examine the effects of selective variations in the model inputs.risk, probability, railroads, hazardous

Optimal Price and Protection Period Decisions for a Product Under Warranty

This paper proposes and analyzes a model for maximizing the profit of a product sold under warranty. The decision variables assumed in the model are the price of the product and the length of the period throughout which the manufacturer is responsible for service. The paper focuses on the dependence of optimal profit on price and protection period for a realistic and general description of a warrantied product. Failures are stochastic and repairs are of constant cost. Demand depends exponentially on price and warranty duration. After an introduction and literature review, the basic model is presented and assumptions are made about the particular probability law governing product failures. The optimization procedure is then illustrated for certain values of input parameters. Finally, an extensive economic analysis of the sensitivity of the optimal solution to variations in input parameters is provided.

Optimal vendor selection in a multiproduct supply chain with truckload discounts

When products are sold by multiple vendors in various locations, the purchaser must decide what to order from each vendor and where to send it. To solve this decision problem, a novel optimization model is developed and applied to a situation involving the nationwide wholesale distribution of grocery products. Comparing the model's solution with the actual record of shipments reveals instances in which the model selected higher-priced vendors in order to capitalize on truckload cost savings, which are seen to be an important factor in vendor selection. Additional models are developed to reduce computation time and assign shipments to vehicles.Vendor selection Supply chain Truckload discount Mixed-integer programming Distribution

Large-scale network distribution of pooled empty freight cars over time, with limited substitution and equitable benefits

A set of models and procedures is described for finding the optimal distribution of empty freight cars owned by the railroads participating in a pooling agreement. A distinction is drawn between a system focus, in which the emphasis is on minimizing total cost, and a company focus, in which the benefits of the agreement to the individual railroads are emphasized. Limited car substitution is accounted for by combining interchange costs with distribution costs, and incorporating interchange possibilities and prohibitions into the network structure. Temporal variations in car supply and demand levels are also taken into account. A large-scale network algorithm is used in conjunction with decomposition to obtain solutions which show for a given time horizon how much equity can be achieved in the balance of savings among the railroads involved and at what cost. Results using actual operating data are reported.

Optimal Allocation of Risk-Reduction Resources in Event Trees

In this paper, we present a novel quantitative analysis for the strategic planning decision problem of allocating certain available prevention and protection resources to, respectively, reduce the failure probabilities of system safety measures and the total expected loss from a sequence of events. Using an event tree optimization approach, the resulting risk-reduction scenario problem is modeled and then reformulated as a specially structured nonconvex factorable program. We derive a tight linear programming relaxation along with related theoretical insights that serve to lay the foundation for designing a tailored branch-and-bound algorithm that is proven to converge to a global optimum. Computational experience is reported for a hypothetical case study, as well as for several realistic simulated test cases, based on different parameter settings. The results on the simulated test cases demonstrate that the proposed approach dominates the commercial software BARON v7.5 when the latter is applied to solve the original model by more robustly yielding provable optimal solutions that are at an average of 16.6% better in terms of objective function value; and it performs competitively when both models are used to solve the reformulated problem, particularly for larger test instances.risk management, risk reduction, event trees, system safety, global optimization, factorable programming, branch-and-bound