Linear Programming LP

Much more common in applications to problems in chemical processing is the use of numerical methods for either nonlinear or linear problems. These methods, which are covered in the following sections of this chapter, are mostly search methods that start from an assumed solution for d and then move d in a series of iterations, by some strategy, to reduce (increase) the objective function to achieve a minimum (maximum). 

Additional complications can be present in optimization problems. For example, the objective function and/or one or more equality constraints may be discontinuous. This might occur, for example, when steam is available to a process at two or three pressure levels, with a different cost at each level. When steam is used where pressure is a variable, the steam cost could change abruptly at a certain value of the pressure, causing a discontinuity in the objective function. 

Let Vyi = gallons of Beer A, Vg = gallons of Beer B, and Vw = gallons of water. The optimization problem is stated as follows 

The problem consists of three variables, two equality constraints, and three lower bounds. Hie problem can be reduced to two decision variables by solving Eq. (18.11) for Vb, 

at the lower limit of Vw, Vg = 37.5. Thus, one vertex is at Vg = 37.5 and V = 0. The other vertex is at the intersection of two lower bounds (V = 10, V = 0). One might argue that another vertex exists at an upper bound on the volume of Beer A, corresponding to an upper bound on the volume of water. This occurs where the blend contains only Beer A and water (V = 88.89, V = 11.11). Now, evaluate the cost at each of these three vertices. The results are 

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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 ... 

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) ... 

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. 

Convex Cases of NLP Problems Linear programs and quadratic programs are special cases of (3-85) that allow for more efficient solution, based on application of KKT conditions (3-88) through (3-91). Because these are convex problems, any locally optimal solution is a global solution. In particular, if the objective and constraint functions in (3-85) are linear, then the following linear program (LP)... 

This chapter focuses on a new approach that allows for the comprehensive planning and optimization of multi-stage production processes - the quant-based combinatorial optimization. First, a distinction is drawn between classical approaches such as Linear Programming (LP) and the quant-based combinatorial approach. Before going into the special characteristics and requirements of the process industry the one model approach with quant-based combinatorial optimization is introduced. Then we will give two examples of how this new approach is applied to real life problems. 

The optimization of value-added processes is a subject that scientists all over the world have been dealing with for more than 70 years. The first basic algorithms for so-called Linear Programming (LP) were developed at American and European universities already in the 1930s, for the first time allowing the planning and simulation of simple business processes. LP soon became the base of the first software systems and even today almost all Supply Chain Management (SCM) or... 

Linear Programming (LP) for continuous variables based on the SIMPLEX algorithm... 

Probably the most prominent approach to large-scale metabolic networks is constraint-based flux balance analysis. The steady-state condition Eq. (63) defines a linear equation with respect to the feasible flux distributions v°. Formulating a set of constraints and a linear objective function, the properties of the solution space P can be exploredusing standard techniques of linear programming (LP). In this case, the flux balance approach takes the form ... 

This example illustrates the planning of petroleum refinery operations as an ordinary single objective linear programming (LP) problem of total daily profit maximization. 

We demonstrate the implementation of the proposed stochastic model formulations on the refinery planning linear programming (LP) model explained in Chapter 2. The original single-objective LP model is first solved deterministically and is then reformulated with the addition of the stochastic dimension according to the four proposed formulations. The complete scenario representation of the prices, demands, and yields is provided in Table 6.2. 

The following linear program (LP) can be formulated to check HEN feasibility at specified values of the supply temperatures ... 

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