Linear programming minimal models

The above formulation is an extension of the deterministic model explained in Chapter 5. We will mainly explain the stochastic part of the above formulation. The above formulation is a two-stage stochastic mixed-integer linear programming (MILP) model. Objective function (9.1) minimizes the first stage variables and the penalized second stage variables. The production over the target demand is penalized as an additional inventory cost per ton of refinery and petrochemical products. Similarly, shortfall in a certain product demand is assumed to be satisfied at the product spot market price. The recourse variables V [ +, , V e)+ and V e[ in... 

To determine the best weights for the weighted moving average method, a linear programming (LP) model can be used. The objective of fhe LP model is to find fhe opfimal weighfs fhaf minimize the forecast error. Let denote the forecast error in period t. Then, e, is given by... 

The minimization of the quadratic performance index in Equation (16.2), subject to the constraints in Equations (16.5-16.7) and the step response model such as Equation (16.1), forms a standard quadratic programming (QP) problem, described in Chapter 8. If the quadratic terms in Equation (16.2) are replaced by linear terms, a linear programming program (LP) problem results that can also be solved using standard methods. The MPC formulation for SISO control problems described earlier can easily be extended to MIMO problems and to other types of models and objective functions (Lee et al., 1994). Tuning the controller is carried out by adjusting the following parameters ... 

K.L. Croxton and B. Gendron and T.L. Magnanti, A comparison of mixed-integer programming models for nonconvex piecewise linear cost minimization problems. Management Science, 49, 9 (2003) 1268-1273... 

The methodologies used for fitting calibration curves depend on whether they are linear or nonlinear. Model fitting basically consists of finding values of the model parameters that minimize the deviation between the fitted curve and the observed data (i.e., to get the curve to fit the data as perfectly as possible). For linear models, estimates of the parameters such as the intercept and slope can be derived analytically. However, this is not possible for most nonlinear models. Estimation of the parameters in most nonlinear models requires computer-intensive numerical optimization techniques that require the input of starting values by the user (or by an automated program), and the final estimates of the model parameters are determined based on numerically optimizing the closeness of the fitted calibration curve to the observed measurements. Fortunately, this is now automated and available in most user-friendly software. 

The residual terms associated with each system of equations represent the difference between the linear programming estimate and the actual concentration of each organic contaminant in the sample. The optimum solution for each system of equations is that for which the residual terms are minimized. Since a perfect modeling solution would accoimt for 100% of the measured concentration for each organic contaminant, the validity of the present environmental forensic MM model can be evaluated by calculating a mean residual percent of each contaminant (the mean residual for each contaminant divided by the mean contaminant concentration) [Ij. Therefore, the use of linear programming technique partitioning helps correct the initial end-member compositions of SWMs and/or their leachates, and their abundances, to better fit the observed multivariate data set, as well as to specify and select the compositions of the end-members. 

Locating a new facility to minimize the total weighted distance from the facility to customers is an easy to solve problem assuming rectilinear distance values. This problem can be modeled as a linear programming problem. Linear programming (LP) problems are easy and efficient algorithms exist that solve them optimally. 

Abstract. In the present paper the problem of reuse water networks (RWN) have been modeled and optimized by the application of a modified Particle Swarm Optimization (PSO) algorithm. A proposed modified PSO method lead with both discrete and continuous variables in Mixed Integer Non-Linear Programming (MINLP) formulation that represent the water allocation problems. Pinch Analysis concepts are used jointly with the improved PSO method. Two literature problems considering mono and multicomponent problems were solved with the developed systematic and results has shown excellent performance in the optimality of reuse water network synthesis based on the criterion of minimization of annual total cost. 

Single objective models In Section 6.2, we presented a single objective linear programming model for order allocation using Example 6.3. The model considered supplier capacities, price discounts, buyers demand, quality, and lead-time constraints. The objective was to minimize total cost, which included fixed and variable cost of the suppliers. We shall briefly review here, some of the other single objective models that have been discussed in the literature. 

Solve a dual problem of the linear programming model obtained by fixing the integer variables and update the upper bound (for minimization problems). 

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