## Introduction to Applied OptimizationOptimization has pervaded all spheres of human endeavor. Although op- mization has been practiced in some form or other from the early prehistoric era, this area has seen progressive growth during the last ?ve decades. M- ern society lives not only in an environment of intense competition but is also constrained to plan its growth in a sustainable manner with due concern for conservation of resources. Thus, it has become imperative to plan, design, operate, and manage resources and assets in an optimal manner. Early - proaches have been to optimize individual activities in a standalone manner, however,thecurrenttrendistowardsanintegratedapproach:integratings- thesis and design, design and control, production planning, scheduling, and control. The functioning of a system may be governed by multiple perf- mance objectives. Optimization of such systems will call for special strategies for handling the multiple objectives to provide solutions closer to the systems requirement. Uncertainty and variability are two issues which render op- mal decision making di?cult. Optimization under uncertainty would become increasingly important if one is to get the best out of a system plagued by uncertain components. These issues have thrown up a large number of ch- lenging optimization problems which need to be resolved with a set of existing and newly evolving optimization tools. Optimization theory had evolved initially to provide generic solutions to optimizationproblemsinlinear,nonlinear,unconstrained,andconstrained- mains. Theseoptimization problems wereoften called mathematical progr- mingproblemswithtwodistinctiveclassi?cations,namelylinearandnonlinear programming problems. |

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amount of frit batch distillation binary variables blending problem Branch-and-bound calculated calculus of variations component computational configuration constraint violations convex decision variables deterministic differential equations discrete distribution Diwekar dynamic programming equality constraints Example expected value feasible region formulation frit mass genetic algorithms given Hessian initial Insulin Time Insulin integer isoperimetric problem iteration linear programming lower bound mass fraction mathematical Maximize maximum principle MILP Minimize minimum MINLP multiobjective optimization multiple Node nondominated set nonlinear programming Nsamp number of samples objective function obtained Operations Research optimal control problems optimal solution optimization problem optimum parameter Pareto set probabilistic procedure random ratio relative volatility representation represented shown in Figure simplex method simplex tableau simulated annealing solve stochastic annealing stochastic optimization stochastic programming strategy Table tank techniques temperature tion uncertain variable uncertainty variance vector waste blend waste composition x1 and x2 zero