Operations Research (OR) is the science dealing with the application of advanced analytics to support decisions and generally the governance of processes and systems. It permits to find the optimal solution for a wide range of problem. For example to calculate the optimal routing of vehicles, to estimate the optimal planning of resources, to define the optimal quantity for the inventory, to calculate the optimal sales prices of products etc.
According to the Oxford English Dictionary, Optimization is the action of making the best or most effective use of a situation or resource. This definition satisfies the common sense, but it shades the much richer contents that we attain when, by adding the adjective Mathematical, we deal with Mathematical Optimization. Indeed Mathematical Optimization is the deep and wide scientific technique aiming to put in practice what the Oxford English Dictionary defines as Optimization, in the real contexts of Engineering, Economics and other Applied Sciences.
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A Mathematical Optimization problem is given in terms of three mathematical objects:
a set of decision variables: these are the variables to which we have to give a value. Usually they correspond to unities of resources, such us money, people, commodities
a set of constraint functions that the decision variables must satisfy: usually given in terms of inequalities when the constraint denotes a limitation on the availability of a resource, given in terms of equalities when the constraint denotes a link among variable
an objective function: it is the performance index associated to the values of the decision variables.
If the decision variables satisfy all the constraint functions, we say that the variables are feasible. The feasible set is the set of all feasible decision variables. Then the mathematical optimization problem can be stated as follows:
among all decision variables belonging to the feasible set, determine the ones that maximize or minimize the objective function.