shabbir ahmed - Academia.edu (original) (raw)

Papers by shabbir ahmed

International Journal of Hydrogen Energy, 2004

Catalytic autothermal reforming is considered one of the most effective methods of producing hydr... more Catalytic autothermal reforming is considered one of the most effective methods of producing hydrogen from heavy hydrocarbon fuels, such as diesel fuel, for fuel cell or emissions reduction applications. This article describes an investigation of the reactor characteristics and ...

European Journal of Operational Research, 2007

We analyze an extension of the classical multi-period, single-item, linear cost inventory problem... more We analyze an extension of the classical multi-period, single-item, linear cost inventory problem where the objective function is a coherent risk measure. Properties of coherent risk measures allow us to offer a unifying treatment of risk averse and min-max type formulations. For the single period newsvendor problem, we show that the structure of the optimal solution of the risk averse model is similar to that of the classical expected value problem. For a finite horizon dynamic inventory model, we show that, again, the optimal policy has a similar structure as that of the expected value problem. This result carries over even to the case when there is a fixed ordering cost. We also analyze monotonicity properties of the optimal order quantity with respect to the degree of risk aversion for certain risk measures.

We study approximations of optimization problems with probabilistic constraints in which the orig... more We study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample. We show that such a sample approximation problem with risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast. This leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound. We then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability. Once again, we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem. Finally, we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints.

Linear programs with joint probabilistic constraints (PCLP) are known to be highly intractable du... more Linear programs with joint probabilistic constraints (PCLP) are known to be highly intractable due to the non-convexity of the feasible region. We consider a special case of PCLP in which only the right-hand side is random and this random vector has a finite distribution. We present a mixed integer programming formulation and study the relaxation corresponding to a single row of the probabilistic constraint, yielding two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. We present computational results that indicate that by using our strengthened formulations, large scale instances can be solved to optimality.

Informs Journal on Computing, 2008

1. Integer feasible solutions might not be in C. 2. Fathoming by integrality might fathom optimal... more 1. Integer feasible solutions might not be in C. 2. Fathoming by integrality might fathom optimal solution.

This paper develops a solution strategy for two-stage stochastic programs with integer recourse. ... more This paper develops a solution strategy for two-stage stochastic programs with integer recourse. The proposed methodology relies on approximating the underlying stochastic program via sampling, and solving the approximate problem via a specialized optimization algorithm. We show that the proposed scheme will produce an optimal solution to the true problem with probability approaching one exponentially fast as the sample size is increased. For fixed sample size, we describe statistical and deterministic bounding techniques to validate the quality of a candidate optimal solution. Preliminary computational experience with the method is reported.

Mathematical Programming, 2006

This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitate... more This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical ( , S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the ( , S) inequalities to a general class of valid inequalities, called the (Q, S Q ) inequalities, and we establish necessary and sufficient conditions which guarantee that the (Q, S Q ) inequalities are facet-defining. A separation heuristic for (Q, S Q ) inequalities is developed and incorporated into a branch-and-cut algorithm. A computational study verifies the usefulness of the (Q, S Q ) inequalities as cuts.

Mathematical Programming, 2010

Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the fe... more Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the right-hand side is random and this random vector has a finite distribution. We give a mixed-integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint. We obtain two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. We present computational results which indicate that by using our strengthened formulations, instances that are considerably larger than have been considered before can be solved to optimality.

European Journal of Operational Research, 2005

This paper proposes a stochastic programming model and solution algorithm for solving supply chai... more This paper proposes a stochastic programming model and solution algorithm for solving supply chain network design problems of a realistic scale. Existing approaches for these problems are either restricted to deterministic environments or can only address a modest number of scenarios for the uncertain problem parameters. Our solution methodology integrates a recently proposed sampling strategy, the Sample Average Approximation scheme, with an accelerated Benders decomposition algorithm to quickly compute high quality solutions to large-scale stochastic supply chain design problems with a huge (potentially infinite) number of scenarios. A computational study involving two real supply chain networks are presented to highlight the significance of the stochastic model as well as the efficiency of the proposed solution strategy. of supply chain networks (see for example ). However, the majority of this research assumes that the operational characteristics of, and hence the design parameters for, the supply chain are deterministic. Unfortunately, critical parameters such as customer demands, prices, and resource capacity are quite uncertain. Moreover, the arrival of regional economic alliances, for instance the Asian Pacific Economic Alliance and the European Union, have prompted many corporations to move more and more towards global supply chains, and therefore to become exposed to risky factors such as exchange rates, reliability of transportation channels, and transfer prices . Unless the supply chain is designed to be robust with respect to the uncertain operating conditions, the impact of operational inefficiencies such as delays and disruptions will be larger than necessary. A recent study found that after a company announces a supply chain disruption, such as a production or shipment delay, its stock price can decrease significantly, with an average decrease of 8.6% on the day of the announcement, and is often followed by further decreases, as much as 20% over the next six months.

Siam Journal on Optimization, 2004

For a particular class of minimax stochastic programming models, we show that the problem can be ... more For a particular class of minimax stochastic programming models, we show that the problem can be equivalently reformulated into a standard stochastic programming problem. This permits the direct use of standard decomposition and sampling methods developed for stochastic programming. We also show that this class of minimax stochastic programs subsumes a large family of mean-risk stochastic programs where risk is measured in terms of deviations from a quantile.

Computational Optimization and Applications, 2003

The sample average approximation (SAA) method is an approach for solving stochastic optimization ... more The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps. We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 21694 scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.