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  • 1
    Electronic Resource
    Electronic Resource
    Sublinear upper bounds for stochastic programs with recourse (1989)
    Springer
    Mathematical programming 43 (1989), S. 131-149 
    Details
    ISSN: 1436-4646
    Keywords: Stochastic programming ; sublinear function ; simple recourse problem ; recourse model ; duality ; approximation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program. The resulting problem's objective involves the inf-convolution of convex functions. A dual of this problem is formulated to obtain an implementable procedure to calculate the bound. Function evaluations for the resulting convex program only require a small number of single integrations in contrast with previous upper bounds that require a number of function evaluations that grows exponentially in the number of random variables. The sublinear bound can often be used when other suggested upper bounds are intractable. Computational results indicate that the sublinear approximation provides good, efficient bounds on the stochastic program objective value.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582286
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  • 2
    Electronic Resource
    Electronic Resource
    The value of the stochastic solution in stochastic linear programs with fixed recourse (1982)
    Springer
    Mathematical programming 24 (1982), S. 314-325 
    Details
    ISSN: 1436-4646
    Keywords: Stochastic Programming ; Linear Programming ; Expected Value of Perfect Information
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Stochastic linear programs have been rarely used in practical situations largely because of their complexity. In evaluating these problems without finding the exact solution, a common method has been to find bounds on the expected value of perfect information. In this paper, we consider a different method. We present bounds on the value of the stochastic solution, that is, the potential benefit from solving the stochastic program over solving a deterministic program in which expected values have replaced random parameters. These bounds are calculated by solving smaller programs related to the stochastic recourse problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585113
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  • 3
    Electronic Resource
    Electronic Resource
    Duality Gaps in Stochastic Integer Programming (2000)
    Springer
    Journal of global optimization 18 (2000), S. 189-194 
    Details
    ISSN: 1573-2916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this note, we explore the implications of a result that suggests that the duality gap caused by a Lagrangian relaxation of the nonanticipativity constraints in a stochastic mixed integer (binary) program diminishes as the number of scenarios increases. By way of an example, we illustrate that this is not the case. In general, the duality gap remains bounded away from zero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008314824754
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  • 4
    Electronic Resource
    Electronic Resource
    Parallel decomposition of large-scale stochastic nonlinear programs (1996)
    Springer
    Annals of operations research 64 (1996), S. 39-65 
    Details
    ISSN: 1572-9338
    Keywords: Decomposition ; economics ; environment ; parallel computation ; stochastic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Many practical decision problems involve both nonlinear relationships and uncertainties. The resulting stochastic nonlinear programs become quite difficult to solve as the number of possible scenarios increases. In this paper, we provide a decomposition method for problems in which nonlinear constraints appear within periods. We also show how the method extends to lower bounding refinements of the set of scenarios when the random data are independent from period to period. We then apply the method to a stochastic model of the U.S. economy based on the Global 2100 method developed by Manne and Richels.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02187640
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  • 5
    Electronic Resource
    Electronic Resource
    Setting Single-Period Optimal Capacity Levels and Prices for Substitutable Products (1998)
    Springer
    International journal of flexible manufacturing systems 10 (1998), S. 407-430 
    Details
    ISSN: 1572-9370
    Keywords: substitutable products ; capacity ; determination
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract In this paper, we consider how a company that has the flexibility to produce two substitutable products would determine optimal capacity levels and prices for these products in a single-period problem. We first consider the case where the firm is a price taker but can determine optimal capacity levels for both products. We then consider the case where the firm can set the price for one product and the optimal capacity level for the other. Finally, we consider the case where capacity is fixed for both products, but the firm can set prices. For each case, we examine the sensitivity of optimal prices and capacities to the problem parameters. Finally, we consider the case where each product is managed by a product manager trying to maximize individual product profits rather than overall firm profits and analyze how optimal price and capacity decisions are affected.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008061605260
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  • 6
    Electronic Resource
    Electronic Resource
    Bounding separable recourse functions with limited distribution information (1991)
    Springer
    Annals of operations research 30 (1991), S. 277-298 
    Details
    ISSN: 1572-9338
    Keywords: Integration ; stochastic programming ; moment problem ; duality ; approximation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract The recourse function in a stochastic program with recourse can be approximated by separable functions of the original random variables or linear transformations of them. The resulting bound then involves summing simple integrals. These integrals may themselves be difficult to compute or may require more information about the random variables than is available. In this paper, we show that a special class of functions has an easily computable bound that achieves the best upper bound when only first and second moment constraints are available.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02204821
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  • 7
    Electronic Resource
    Electronic Resource
    Continuous approximation schemes for stochastic programs (1995)
    Springer
    Annals of operations research 56 (1995), S. 15-38 
    Details
    ISSN: 1572-9338
    Keywords: Approximation ; derivative ; continuous distribution ; stochastic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract One of the main methods for solving stochastic programs is approximation by discretizing the probability distribution. However, discretization may lose differentiability of expectational functionals. The complexity of discrete approximation schemes also increases exponentially as the dimension of the random vector increases. On the other hand, stochastic methods can solve stochastic programs with larger dimensions but their convergence is in the sense of probability one. In this paper, we study the differentiability property of stochastic two-stage programs and discuss continuous approximation methods for stochastic programs. We present several ways to calculate and estimate this derivative. We then design several continuous approximation schemes and study their convergence behavior and implementation. The methods include several types of truncation approximation, lower dimensional approximation and limited basis approximation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02031698
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  • 8
    Electronic Resource
    Electronic Resource
    Models and model value in stochastic programming (1995)
    Springer
    Annals of operations research 59 (1995), S. 1-18 
    Details
    ISSN: 1572-9338
    Keywords: Stochastic optimization models ; finance ; quality control ; engineering design
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Finding optimal decisions often involves the consideration of certain random or unknown parameters. A standard approach is to replace the random parameters by the expectations and to solve a deterministic mathematical program. A second approach is to consider possible future scenarios and the decision that would be best under each of these scenarios. The question then becomes how to choose among these alternatives. Both approaches may produce solutions that are far from optimal in the stochastic programming model that explicitly includes the random parameters. In this paper, we illustrate this advantage of a stochastic program model through two examples that are representative of the range of problems considered in stochastic programming. The paper focuses on the relative value of the stochastic program solution over a deterministic problem solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02031741
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  • 9
    Electronic Resource
    Electronic Resource
    Stochastic programming approaches to stochastic scheduling (1996)
    Springer
    Journal of global optimization 9 (1996), S. 417-451 
    Details
    ISSN: 1573-2916
    Keywords: stochastic programming ; hierarchical decision making ; stochastic scheduling ; queueing networks ; large deviations ; turnpikes ; Lagrangian relaxation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Practical scheduling problems typically require decisions without full information about the outcomes of those decisions. Yields, resource availability, performance, demand, costs, and revenues may all vary. Incorporating these quantities into stochastic scheduling models often produces diffculties in analysis that may be addressed in a variety of ways. In this paper, we present results based on stochastic programming approaches to the hierarchy of decisions in typical stochastic scheduling situations. Our unifying framework allows us to treat all aspects of a decision in a similar framework. We show how views from different levels enable approximations that can overcome nonconvexities and duality gaps that appear in deterministic formulations. In particular, we show that the stochastic program structure leads to a vanishing Lagrangian duality gap in stochastic integer programs as the number of scenarios increases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00121682
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  • 10
    Electronic Resource
    Electronic Resource
    A startup procedure for process industries using a multiple objective nonlinear program (1998)
    Springer
    IIE transactions 30 (1998), S. 291-300 
    Details
    ISSN: 1573-9724
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract We develop a multiple objective nonlinear program to determine optimal set points for the startup period of a continuous mix manufacturing process. We model the objectives and constraints dictated by the problem formulation and solve the model. We discuss why and how process adjustments are made at the startup of operations. To advance the model, we develop a startup adjustment methodology. This methodology guides the process for startup conditions and real-time operation of the system. It is demonstrated on data from a breakfast cereal manufacturing process where the startup process under conventional approaches spans several days. We close the paper with discussions on process adjustments after steady state has been achieved and strategic issues in process regulation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007438822496
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