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In this paper, we develop new discrete relaxations for nonlinear expressions in factorable programming. We utilize specialized convexification results as well as composite relaxations to develop mixed-integer programming relaxations. Our relaxations rely on ideal formulations of convex hulls of outer-functions over a combinatorial structure that captures local inner-function structure. The resulting relaxations often require fewer variables and are tighter than currently prevalent ones. Finally, we provide computational evidence to demonstrate that our relaxations close approximately 60\%--70\% of the gap relative to McCormick relaxations and significantly improve the relaxations used in a state-of-the-art solver on various instances involving polynomial functions. [ABSTRACT FROM AUTHOR]

We study the pricing of cardinality bundles, where firms set prices that depend only on the size of the purchased bundle, a practice that is increasingly being adopted by industry. The model we study, where consumer choices are discrete, was originally proposed by Hitt and Chen [Hitt L, Chen P (2005) Bundling with customer self-selection: A simple approach to bundling low-marginal-cost goods. Management Sci. 51(10):1481-1493], and it requires that consumers' preferences obey the Spence-Mirrlees single-crossing property. We correct prior approaches and develop various structural and managerial insights. We develop a fast combinatorial technique to obtain the optimal prices. We extend our analysis to address a quantity discount problem originally proposed in Spence [Spence M (1980) Multi-product quantity-dependent prices and profitability constraints. Rev. Econom. Stud. 47(5):821-841j. We provide examples that demonstrate that the proposed approach of Spence (1980) only identifies local optima without providing guidance on selecting the globally optimal pricing function. Our insights from the discrete model are extended to this context to develop a scheme that provides solutions within an arbitrary prespecified tolerance. Consequently, we also solve the continuous version of the cardinality bundling problem. History: Accepted by Chris Forman, information systems. Keywords: cardinality bundling * marketing * economics of information systems * pricing, 1. Introduction Bundling and its benefits have been studied extensively in the literature. For example, Bakos and Brynjolfsson (1999) show that when products are synergistic, offering bundles of products can [...]

Separation of mixtures of chemicals, ubiquitous in chemical and petrochemical industries, by distillation is energy intensive. Nearly 3% of the overall energy is used for distillation in the United States. Improving the distillation process is crucial for making chemical industries more sustainable. However, designing distillation sequences is challenging because the choice set is vast, and the equations governing the physical process are highly nonconvex. Traditional design practices rely on heuristics and often result in suboptimal solutions. Tumbalam Gooty et al. present the first approach that reliably identifies the distillation sequence that requires the least energy for a given separation. By embedding convex hulls of substructures and adapting the reformulation-linearization technique to fractions of polynomials, they demonstrated that their approach outperforms the state-of-the-art. Their work will help the chemical industry reduce greenhouse gas emissions associated with distillation. In this paper, we describe the first mixed-integer nonlinear programming (MINLP)-based solution approach that successfully identifies the most energy-efficient distillation configuration sequence for a given separation. Current sequence design strategies are largely heuristic. The rigorous approach presented here can help reduce the significant energy consumption and consequent greenhouse gas emissions by separation processes. First, we model discrete choices using a formulation that is provably tighter than previous formulations. Second, we highlight the use of partial fraction decomposition alongside reformulation-linearization technique (RLT). Third, we obtain convex hull results for various special structures. Fourth, we develop new ways to discretize the MINLP. Finally, we provide computational evidence to demonstrate that our approach significantly outperforms the state-of-the-art techniques. Funding: This work was supported by the U.S. Department of Energy [Grant DE-EE0005768], the Bilsland Dissertation Fellowship, and the National Science Foundation [Grant EEC-1647722]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/opre.2022.2340. [ABSTRACT FROM AUTHOR]

The constant molar overflow (CMO) framework, while useful for shortcut distillation models, assumes that all components have the same latent heats of vaporization. A simple transformation, from molar flows to latent‐heat flows, allows shortcut models to retain the mathematical simplicity of the CMO framework while accounting for different latent heats, resulting in the constant heat transport (CHT) framework for adiabatic distillation columns. Although several past works have already proposed this transformation in the literature, it has not been well utilized in recent times. In this article, we show the utility of this transformation in upgrading various applications such as identifying energy‐efficient multicomponent distillation configurations based on heat duty rather than surrogate vapor flow. The method transforms the Vmin diagram to a Qmin diagram. Furthermore, we derive new and insightful analytical results in distillation, such as cumulative latent‐heat stage fractions having monotonic profiles within a distillation column under the CHT framework. [ABSTRACT FROM AUTHOR]

Multi‐feed, multi‐product distillation columns are ubiquitous in multicomponent distillation systems. The minimum reflux ratio of a distillation column is directly related to its energy consumption and capital cost. Thus, it is a key parameter for distillation systems design, operation, and comparison. In this series, we present the first accurate shortcut based algorithmic method to determine the minimum reflux condition for any general multi‐feed, multi‐product (MFMP) distillation column separating any ideal multicomponent mixture. The classic McCabe‐Thiele or Underwood method is a special case of this general approach. Compared with existing techniques, this method does not involve any rigorous tray‐by‐tray calculation, nor does it require guessing of key components. In this first part of the series, we present the mathematical model for a general MFMP column, derive constraints for feasible separation and minimum reflux condition, discuss their geometric interpretations, and present an illustrative example to demonstrate the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

We develop techniques to convexify a set that is invariant under permutation and/or change of sign of variables and discuss applications of these results. First, we convexify the intersection of the unit ball of a permutation and sign-invariant norm with a cardinality constraint. This gives a nonlinear formulation for the feasible set of sparse principal component analysis (PCA) and an alternative proof of the K-support norm. Second, we characterize the convex hull of sets of matrices defined by constraining their singular values. As a consequence, we generalize an earlier result that characterizes the convex hull of rank-constrained matrices whose spectral norm is below a given threshold. Third, we derive convex and concave envelopes of various permutation-invariant nonlinear functions and their level sets over hypercubes, with congruent bounds on all variables. Finally, we develop new relaxations for the exterior product of sparse vectors. Using these relaxations for sparse PCA, we show that our relaxation closes 98% of the gap left by a classical semidefinite programming relaxation for instances where the covariance matrices are of dimension up to 50 × 50. [ABSTRACT FROM AUTHOR]

We systematically analyze power requirements of membrane and distillation processes for binary mixtures where the desired product component is more permeable and also more volatile. We first derive a shortcut method to compare the efficiency of heat pump and steam-driven distillations. Then, power requirements of heat pump distillation and membrane separation are discussed. Distillation generally requires lower power when either high component recoveries are needed (at all tested product purities), or high purity product streams with modest recoveries are needed. For high purity products at modest recoveries, membranes have a potential to provide energy benefits for highly enriched feeds, especially those composed of close boiling components. Additionally, when feed concentration is moderate to high and product recovery and purity are modest, membranes are likely to show efficiency gain. For the advantageous distillation scenarios studied, the power was generally lower than the membranes by a factor of two to seven. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new relaxations for the hypograph of composite functions assuming that the outer function is supermodular and concave extendable. Relying on a recently introduced relaxation framework, we devise a separation algorithm for the graph of the outer function over P, where P is a special polytope to capture the structure of each inner function using its finitely many bounded estimators. The separation algorithm takes O (d n log d) time, where d is the number of inner functions and n is the number of estimators for each inner function. Consequently, we derive large classes of inequalities that tighten prevalent factorable programming relaxations. We also generalize a decomposition result and devise techniques to simultaneously separate hypographs of various supermodular, concave-extendable functions using facet-defining inequalities. Assuming that the outer function is convex in each argument, we characterize the limiting relaxation obtained with infinitely many estimators as the solution of an optimal transport problem. When the outer function is also supermodular, we obtain an explicit integral formula for this relaxation. [ABSTRACT FROM AUTHOR]

The article discusses that commercial process simulators software has become standard in industry and an integral part of the undergraduate chemical engineering curriculum. It mentions that shortcut methods are useful for conducting an initial screen of available configuration options. It also mentions that relative volatility is a measure of the distribution of components between the liquid and vapor phases of a mixture at equilibrium.

Heat and mass integration to consolidate distillation columns in a multicomponent distillation configuration can lead to a number of new energy efficient and cost‐effective configurations. In this work, a powerful and simple‐to‐use fact about heat and mass integration is identified. The newly developed heat and mass integrated configurations, which we call as HMP configurations, involve first introducing thermal couplings to all intermediate transfer streams, followed by consolidating columns associated with a lighter pure product reboiler and a heavier pure product condenser. A systematic method of enumerating all HMP configurations is introduced. The energy savings of HMP configurations is compared with the well‐known fully thermally coupled (FTC) configurations. HMP configurations can have very similar and sometimes even the same minimum total vapor duty requirement as the FTC configuration is demonstrated, while using far less number of column sections, intermediate transfer streams, and thermal couplings than the FTC configurations. © 2018 American Institute of Chemical Engineers AIChE J, 64: 3410–3418, 2018 [ABSTRACT FROM AUTHOR]

We present a simple rule that, for the first time, enables exhaustive enumeration of dividing wall columns (DWCs) corresponding to any given thermally coupled distillation column-configuration. With the successive application of our rule, every partition in a DWC can be extended all the way to the top and/or to the bottom of a column without losing thermodynamic equivalence to the original thermally coupled configuration. This leads to easy-to-operate DWCs with possible control/regulation of each and every vapor split by external means. As a result, we conclude that any given DWC can be transformed into a thermodynamically equivalent form that is easy-to-operate, and hence, there always exists at least one easy-to-operate DWC for any given thermally coupled distillation. Our method of enumerating and identifying easy-to-operate DWCs for an attractive thermally coupled configuration will contribute toward process intensification by providing ways to implement efficient and low-cost multicomponent distillations. © 2017 American Institute of Chemical Engineers AIChE J, 64: 660-672, 2018 [ABSTRACT FROM AUTHOR]

We study the simultaneous convexification of graphs of bilinear functions gk(x; y) = yTAkx over x ∈ ≅ = {x ∈ [0, 1]n ∣Ex ≥ ƒ} and y ∈ Δm = {y ∈ ℝ+m∣ 1Ty ≤ 1}. We propose a constructive procedure to obtain a linear description of the convex hull of the resulting set. This procedure can be applied to derive convex and concave envelopes of certain bilinear functions, to study unary expansions of integer variables in mixed integer bilinear sets, and to obtain convex hulls of sets with complementarity constraints. Exploiting the structure of ≅, the procedure naturally yields stronger linearizations for bilinear terms in a variety of practical settings. In particular, we demonstrate the effectiveness of the approach by strengthening the traditional dual formulation of network interdiction problems and report encouraging preliminary numerical results. [ABSTRACT FROM AUTHOR]

We present a general Global Minimization Algorithm (GMA) to identify basic or thermally coupled distillation configurations that require the least vapor duty under minimum reflux conditions for separating any ideal or near-ideal multicomponent mixture into a desired number of product streams. In this algorithm, global optimality is guaranteed by modeling the system using Underwood equations and reformulating the resulting constraints to bilinear inequalities. The speed of convergence to the globally optimal solution is increased by using appropriate feasibility and optimality based variable-range reduction techniques and by developing valid inequalities. The GMA can be coupled with already developed techniques that enumerate basic and thermally coupled distillation configurations, to provide for the first time, a global optimization based rank-list of distillation configurations. © 2016 American Institute of Chemical Engineers AIChE J, 62: 2071-2086, 2016 [ABSTRACT FROM AUTHOR]