Mixed Integer Linear Optimization Problem Research Articles

Piecewise-Linear Approximations of Multidimensional Functions

We develop explicit, piecewise-linear formulations of functions f(x):ℝ n ↦ℝ, n≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global optimality, then non-analytic functions can be used as an objective or constraint function for large optimization problems. Linear interpolation between fixed gridpoints can also be used to approximate generic, nonlinear functions, allowing us to approximately solve problems using mixed-integer linear optimization methods. Toward this end, we develop two different explicit formulations of piecewise-linear functions and discuss the consequences of integrating the formulations into an optimization problem.

A multiobjective optimization model for processing and distributing biosolids to reuse fields

Advanced wastewater treatment plants remove deleterious nutrients, chemicals, and microorganisms from wastewater and produce biosolids products to be reused at farms and other sites. These biosolids are carefully regulated by environmental restrictions but still may be malodorous to the local populations. In this paper, we develop a multiobjective optimization model to simultaneously minimize the biosolids odours as well as processing and distribution costs. The model employs a linear odour function and bilinear costs; the latter being approximated via Schur's decomposition and special ordered set (SOS) type 2 variables resulting in a mixed integer linear multiobjective optimization problem. Such a model can be used proactively by these plants to produce the least malodorous product at minimal costs. We demonstrate use of the model with a case study for the Blue Plains advanced wastewater treatment plant run by the DC Water and Sewer Authority in Washington, DC.

Including metabolite concentrations into flux balance analysis: thermodynamic realizability as a constraint on flux distributions in metabolic networks.

BackgroundIn recent years, constrained optimization – usually referred to as flux balance analysis (FBA) – has become a widely applied method for the computation of stationary fluxes in large-scale metabolic networks. The striking advantage of FBA as compared to kinetic modeling is that it basically requires only knowledge of the stoichiometry of the network. On the other hand, results of FBA are to a large degree hypothetical because the method relies on plausible but hardly provable optimality principles that are thought to govern metabolic flux distributions.ResultsTo augment the reliability of FBA-based flux calculations we propose an additional side constraint which assures thermodynamic realizability, i.e. that the flux directions are consistent with the corresponding changes of Gibb's free energies. The latter depend on metabolite levels for which plausible ranges can be inferred from experimental data. Computationally, our method results in the solution of a mixed integer linear optimization problem with quadratic scoring function. An optimal flux distribution together with a metabolite profile is determined which assures thermodynamic realizability with minimal deviations of metabolite levels from their expected values. We applied our novel approach to two exemplary metabolic networks of different complexity, the metabolic core network of erythrocytes (30 reactions) and the metabolic network iJR904 of Escherichia coli (931 reactions). Our calculations show that increasing network complexity entails increasing sensitivity of predicted flux distributions to variations of standard Gibb's free energy changes and metabolite concentration ranges. We demonstrate the usefulness of our method for assessing critical concentrations of external metabolites preventing attainment of a metabolic steady state.ConclusionOur method incorporates the thermodynamic link between flux directions and metabolite concentrations into a practical computational algorithm. The weakness of conventional FBA to rely on intuitive assumptions about the reversibility of biochemical reactions is overcome. This enables the computation of reliable flux distributions even under extreme conditions of the network (e.g. enzyme inhibition, depletion of substrates or accumulation of end products) where metabolite concentrations may be drastically altered.

Global pairwise sequence alignment through mixed-integer linear programming: a template-free approach

The problem of protein sequence alignment is the main starting point for biological analysis of genomic information. A new approach to global pairwise sequence alignment is proposed. This approach is formulated in a mathematically rigorous way, as a mixed integer linear optimization (MILP) problem. Not only does the proposed formulation guarantee the identification of the global optimal alignment, but it also allows for a complete rank-ordered list of any number of user specified top ranking pairwise alignments and for the incorporation of restraints based on restrictions necessary to maintain biological function. This approach has been applied to the alignment of transmembrane helices and serine proteases to demonstrate its utility and advantages over other algorithms.

Uniform Marginal Pricing for the Remuneration of Distribution Networks

A new method for distribution access via uniform pricing for the remuneration of distribution networks is presented. The proposed approach merges in a unified framework the investments, the optimal network operation requirements, the effect of the price elasticity of demand, and the application of hourly pricing for demand side management purposes. Hourly uniform marginal prices-understood as tariffs of use of the network-are obtained from maximum social welfare condition sending efficient signals to the utility and consumers, related to the optimal operation of the grid and use of the energy at peak and valley hours. This method is used in the context of a Performance Based Ratemaking regulation to get model companies from operational optimized real networks. Capital fees are integrated in the marginal tariff of use, by means of the New Replacement Value concept, broadly used in yardstick competition. The model is stated as a mixed-integer linear optimization problem suitable to be solved through well-known linear programming tools. The methodology has been successfully tested in a 42-bus test distribution network.

Time-optimal scheduling for high throughput screening processes using cyclic discrete event models

A method for solving the scheduling problem for a class of cyclic systems with respect to throughput maximization is presented. A strictly cyclic mode of operation is considered, where the time offset between the start of consecutive jobs is always constant. All jobs have to follow an identical time scheme. The time scheme may be restricted by due dates or time window constraints. There are no buffers between the resources of the system. Based on discrete events systems (DES) modeling, this job shop scheduling problem can be formulated as a mixed integer linear optimization problem. Throughput is maximized by minimization of the cycle time. The method is applied to high throughput screening (HTS) problems and illustrated by means of a small example.

Modeling and Optimization for High-Throughput-Screening Systems

The problem of cyclic scheduling under the requirement of throughput maximization is considered for a special class of cyclically repeated batch processes. All batches have to follow an identical time scheme. The same resource may be visited more than ouce by the same batch and time window constraints may be stated by the user. It is shown that the cyclic scheduling problem can be transformed into a mixed integer linear optimization problem. The method'sapplication to High-Throughput-Screening processes is demonstrated.

Optimal energy management in pulp and paper mills

In this paper, we examine the utilization of mathematical programming tools for optimum energy management of the power plant in pulp and paper mills. The objective is the fulfillment of the total plant requirements in energy and steam with the minimum possible cost. The proposed methodology is based on the development of a detailed model of the power plant using mass and energy balances and a mathematical formulation of the electrical purchase contract, which can be translated into a rigorous mixed integer linear programming optimization problem. The results show that the method can be a very useful tool for the reduction of production cost due to minimization of the fuel and electricity costs.

Design of Multi-echelon Supply Chain Networks under Demand Uncertainty

We consider the design of multiproduct, multi-echelon supply chain networks. The networks comprise a number of manufacturing sites at fixed locations, a number of warehouses and distribution centers of unknown locations (to be selected from a set of potential locations), and finally a number of customer zones at fixed locations. The system is modeled mathematically as a mixed-integer linear programming optimization problem. The decisions to be determined include the number, location, and capacity of warehouses and distribution centers to be set up, the transportation links that need to be established in the network, and the flows and production rates of materials. The objective is the minimization of the total annualized cost of the network, taking into account both infrastructure and operating costs. A case study illustrates the applicability of such an integrated approach for production and distribution systems with or without product demand uncertainty.

Stability in disjunctive optimization II:continuity of the feasible and optimal set

A disjunctive linear optimization problem is to minimize a function over a feasible set which is the union of (perhaps infinitely many) convex sets. The classofdisjunctiveoptimization problems contains many classes of optimization problems which look at the first sight very different from their structure (f.e. semi-infinite and mixed-integer linear optimization problems, certain non-linear approximation problems, fractional pro¬gramming, linear complementary problems). Our purpose in this paper is to investigate the disjunctive linear optimization problem in dependence on the involved mappings. Especially, we establish necessary and sufficient conditions for the upper and lower semicontinuity of the feasible and optimal set mapping. Examples will illustrate the difficulties and characteristics to derive such conditions in parametric disjunctive programming

Preventive maintenance scheduling of power generating units

The problem of preventive maintenance planning of electric power generating units can be formulated as a mixed-integer linear optimization problem. An extension of the model is presented to deal with more realistic assumptions about utilization of power resource. We propose a heuristic iterative exchange procedure to solve these problems. We introduce two methods to prevent jamming situations outside the feasible domain or at a local optimum. The first method is a recursive exchange procedure called multiple exchanges method and the second relies on Lagrangian relaxation. Furthermore, we compare these procedures with a tabu search.

An MILP formulation for heat‐integrated distillation sequence synthesis

AbstractIn this paper the problem of distillation sequence synthesis with heat integration is formulated and solved as a mixed integer linear superstructure optimization problem. Allowing both conventional and multieffect distillation, two alternatives for generating a superstructure are shown, then a formulation is presented which approximates the synthesis problem as a mixed integer linear programming (MILP) problem. Examples illustrate the method. Reported computer times suggest that earlier algorithmic methods for discovering the better sequences among those allowed by the problem formulation, and without the option of multieffecting, are an order of magnitude slower than this approach.

Die konvexe erweiterungsmethode-eine methode zur lösung diskreter optimierungsaufgaben

For a general discrete optimization problem a sufficient optimality condition is formulated. Starting with this condition a procedure is developped in order to solve the regarded problem. Tho finiteness of the procedure is proved. Furthermore, it is described, how this method can be epplied for the solution of mixed-integer linear optimization problems as well as for mixed linear zero-one problems.