مولود موساوي

Operations research and mathematical optimization When making a decision, one usually needs to consider a large number of alternatives. Each of these must be evaluated using one or several criteria in order to determine the "best" decision. Operational Research is the discipline of developing quantitative tools to assist decision-makers with these often complex decisions. Operational Research will generate user-friendly models representing the real-life situation and its constraints, and will provide a clear formulation of the criteria guiding the choices, and thereby rationalise the decision-making process. A MAJOR ASPECT OF OPERATIONAL RESEARCH - MATHEMATICAL PROGRAMMING Many business decisions - whether strategic and global or operational and local - involve identifying the resources required to best address business needs (such as allocating production tasks to industrial equipment, organising logistics flows, optimising the use of transportation resources for people or merchandise, and assigning shifts to employees). Such decisions often involve a very large number of quantitative initial choices. For instance, in order to create a master production plan we must decide which products are to be manufactured where, when, and in what quantity, and distributed from which site to which customer, when and in what quantity, etc. These choices must be mutually consistent and must meet the constraints specific to the given context. Such decisions can be very difficult to both prepare and assess using traditional methods. In order to make the best possible choice(s) concerning a complex problem, we need to optimise resource allocation according to several variables (criterion, objective function), which themselves are subject to a set of constraints. In mathematical terms, a Mathematical Programming problem can be stated as: Max: f(x1, x2, …, xn) Meeting the constraints (c1, c2, …, cm) Mathematical Programming offers a wide range of algorithms for resolving this type of problem. These methods notably comprise: Linear Programming and its extensions Non-linear Programming Evolutionary algorithms and meta-heuristics Some problems involving multiple criteria will require the enhancement of our optimisation approach via a search for the best compromise. For instance, if two criteria are used to evaluate the problem, a feasibe solution may be selected as optimal when there is no better solution for both criteria simultaneously. For a given problem, there may be many of these feasible solutions, each one proposing a different compromise. In this case, the decision-maker may need to explore some or all of these solutions along the so-called "Pareto frontier". This approach (which is very useful for design optimisation problems, for instance) may require very intensive optimisation calculations entailing the use either of High Performance Computing (HPC) or of evolutionary algorithms such as genetic algorithms. The random nature or inaccuracy of the environment where these solutions must be applied may require a more detailed approach involving Stochastic Programming or Robust Programming methods.