Other Mathematical Programming Models

Multi-objective, stochastic, and nonlinear mathematical programming models are other models that find application in supply chain configuration. 

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All in all, the petroleum industry has invested considerable effort in developing sophisticated mathematical programming models to help planners provide overall planning schemes for refinery operations, crude oil evaluation, and other related tasks. 

Mathematical programming models are used to optimize decisions concerning execution of certain activities subject to resource constraints. Mathematical programming models have a well-defined structure. They consist of mathematical expressions representing objective function and constraints. The expressions involve parameters and decision variables. The parameters are input data, while the decision variables represent the optimization outcome. The objective function represents modeling objectives and makes some decisions more preferable than others. The constraints limit the values that decision variables can assume. 

There are also some important limitations. Mathematical programming models have a lower level of validity compared to some other typies of models—particularly, simulation. In the supply chain configuration context, mathematical programming models have difficulties representing the dynamic and stochastic aspects of the problem. Additionally, solving of many supply chain configuration problems is computationally challenging. 

Prior to the fitting, the chemical reaction model on which the analysis will be based needs to be defined. As mentioned above ReactLab and other modem programs incorporate a model translator that allows the definition in a natural chemistry language and which subsequently translates automatically into internal coefficient information that allows the automatic construction of the mathematical expressions required by the numerical and spieciation algorithms. Note for each reaction an initial guess for the rate constant has to be supplied. The ReactLab model is for this reaction is shown in Figure 9. 

The va/Mg-based approach significantly improves the effectiveness of procedures of controlling chemical reactions. Optimal control on the basis of the value method is widely used with Pontryagin s Maximum Principle, while simultaneously calculating the dynamics of the value contributions of individual steps and species in a reaction kinetic model. At the same time, other methods of optimal control are briefly summarized for a) calculus of variation, b) dynamic programming, and c) nonlinear mathematical programming. 

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