Optimization linear programming

A. M. Morshedi, C. R. Cutier, and T. A. Skrovanek, "Optimal Solution of Dynamic Matrix Control with Linear Programming Techniques,"... 

In real-life problems ia the process iadustry, aeady always there is a nonlinear objective fuactioa. The gradieats deteroiiaed at any particular poiat ia the space of the variables to be optimized can be used to approximate the objective function at that poiat as a linear fuactioa similar techniques can be used to represent nonlinear constraints as linear approximations. The linear programming code can then be used to find an optimum for the linearized problem. At this optimum poiat, the objective can be reevaluated, the gradients can be recomputed, and a new linearized problem can be generated. The new problem can be solved and the optimum found. If the new optimum is the same as the previous one then the computations are terminated. 

Miirty, Linear Programming, Wiley, New York, 1983. Reklaitis, Ravin-dran, and Ragsdell, Engineeiing Optimization, Wiley—Interscience, New York,... 

Production Controls The nature of the produc tion control logic differs greatly between continuous and batch plants. A good example of produc tion control in a continuous process is refineiy optimization. From the assay of the incoming crude oil, the values of the various possible refined products, the contractual commitments to dehver certain products, the performance measures of the various units within a refinery, and the hke, it is possible to determine the mix of produc ts that optimizes the economic return from processing this crude. The solution of this problem involves many relationships and constraints and is solved with techniques such as linear programming. 

Once the highest steam level is set, then intermediate levels must be established. This involves having certain turbines exhaust at intermediate pressures required of lower pressure steam users. These decisions and balances should be done by in-house or contractor personnel having extensive utility experience. People experienced in this work can perform the balances more expeditiously than people with primarily process experience. Utility specialists are experienced in working with boiler manufacturers on the one hand and turbine manufacturers on the other. They have the contacts as well as knowledge of standard procedures and equipment size plateaus to provide commercially workable and optimum systems. At least one company uses a linear program as an aid in steam system optimization. 

Having a linear program available allows the designer to generate better designs, but does not necessarily make his job easier. In fact, it may put additional pressure on him to be sure his results are optimal rather than just a satisfactory design. 

We are now in a position to formulate the problem of minimizing the cost of MSAs. By adopting the linear-programming formulation (P6.1), one can write the following optimization program ... 

The above equations coupled with Eqs. (9.10)-<9.18) represent the constraints of the CHARMEN-synthesis formulation. The objective is to minimize the cost of MSAs and heating/cooling utilities. This is a linear-programming formulation whose solution determines the optimal flowrate and temperature of each substream and heating/cooling utilities. In order to demonstrate this formulation, let us consider the following example. 

These non-linear equations can be embedded into the refinery s linear program (LP) to achieve compliance and optimize the gasoline blend. The key FCC gasoline components that influence RFG are ... 

Another illustration of operations research is the use of linear programming techniques (Section 5.14) to obtain optimal mixtures of gasoline ingredients that will produce a result suitable for different climatic conditions and subject to demand constraints on a long-range basis. 

The last formulation 3rields an analytical method for treating optimal flow. There are special types of linear programming problems (e.g.,... 

Further Comments on General Programming.—This section will utilize ideas developed in linear programming. The use of Lagrange multipliers provides one method for solving constrained optimization problems in which the constraints are given as equalities. 

The problem [Eq. (15)] is a minimax optimization problem. For the case (as it is here) where the approximating function depends linearly on the coefficients, the optimization problem [Eq. (15)] has the form of the Chebyshev approximation problem and has a known solution (Murty, 1983). Indeed, it can be easily shown that with the introduction of the dummy variables z, z, z the minimax problem can be transformed to the following linear program (LP) ... 

Problems resembling the first example, but much more complex, are often studied in industry. For instance in the agro-food industry linear programming is a current tool to optimize the blending of raw materials (e.g. oils) in order to obtain the wanted composition (amount of saturated, monounsaturated and polyunsaturated fatty acids) or property of the final product at the best possible price. Here linear programming is repeatedly applied each time when the price of raw materials is adapted by changing markets. 

Gill, P.E. and W. Murray, "Newton-type Methods for Unconstrained and Linearly Constrained Optimization", Mathematical Programming, 7,311-350 (1974). 

One of the approaches that can be used in design is to carry out structural and parameter optimization of a superstructure. The structural optimization required can be carried out using mixed integer linear programming in the case of a linear problem or mixed integer nonlinear programming in the case of a nonlinear problem. Stochastic optimization can also be very effective for structural optimization problems. 

The linear programming approach outlined in Section IV,B,l,b has also been applied to cyclic networks (Kl), the lack of theoretical validity notwithstanding. On an operational level, linear programming has been used to determine the most efficient means of supplying the water requirements of a major metropolitan area (G3) and to guide the allocation of production and supply of gas for the northwestern counties of England (BIO). In the latter application the results of the grid optimization are used to determine (i) optimal allocation of natural gas supply, (ii) production... 

As shown in Fig. 3-53, optimization problems that arise in chemical engineering can be classified in terms of continuous and discrete variables. For the former, nonlinear programming (NLP) problems form the most general case, and widely applied specializations include linear programming (LP) and quadratic programming (QP). An important distinction for NLP is whether the optimization problem is convex or nonconvex. The latter NLP problem may have multiple local optima, and an important question is whether a global solution is required for the NLP. Another important distinction is whether the problem is assumed to be differentiable or not. 

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