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Typical real world optimization problems arise in the following areas:

Supply Chain Management
Transport and Distribution
Production Planning and Scheduling
Financial and Investment Optimization
Telecommunication Network Design
Portfolio optimization / strategic planning
investment and divestment planning
depot location problem
network and/or site design planning
material flow problems
blending problems
scheduling and timetabling problems
cutting stock and trim loss problems
process design and engineering
power generation

A Survey of Real World Problems

Here we describe in general some areas in which applications of (linear and nonlinear) mixed integer optimization are found. This list is by far not complete.

The brief survey of real world problems given in this section is typical for
many businesses and industries but many other applications also occur in
other areas:

Companies which are in a situation to utilize the advantages of a complex
production network, often because they operate at several sites, may greatly benefit from production planning and production scheduling. Of course, scheduling problems occur also in other branches of industry. These are operational branches and require detailed answers to the questions: when is the production of a specific product on a specific machine to be started? What does the daily production sheet of a worker look like? Scheduling problems belong to a class of the most difficult problems in optimization. Typical special structures, which can be tackled by optimization, are minimal production rates, minimal utilization rates, minimal production or transport amounts. Items of production must also be sequenced.

Cutting stock or trimloss problems occurs in the metal or paper industry. Depending on the cutting machines, one-dimensional or two-dimensional patterns are cut. It is important that the mathematical optimization model and algorithms reflects precisely the cutting logic.

Typical for the chemical process industry, but in modified form also for the
mineral oil or food industry, are blending problems. They occur in a wide variety. In Kallrath (1995) a model is described for finding minimal cost blending which simultaneously includes container handling conditions and other logistic constraints.

In planning it is often required to make a selection from a series of projects in an optimal manner. This is clearly another type of optimization problem.

The question of how a telecommunication network should be structured and
designed when the annual demand is known, or what the traffic infrastructure
should look like for a given traffic demand lead to network design problems.

Specific skills often mean that only certain people can perform specialized
tasks. Thus optimization methods can solve problems of allocating personnel to tasks. Manpower modelling is discussed in Schindler and Semmel (1993). People often need to be timetabled, i.e., allocated to sessions with conflicts being avoided. These applications lead to complex optimization problems. They can be classified as resource constrained scheduling problem with time windows.

Problems of distribution and logistics are fruitful applications for optimization - choosing vehicles, routes, loads and so on. While the problems listed above can be solved with linear mixed-integer methods, problems occurring in process industry very often lead to nonlinear discrete problems. Problems in the refinery or petrcochemical industry include blending problems leading to so-called pooling problems. If, in addition, we have to consider that plants operate in discrete modes, or that connections between tanks and cracker or tanks and vacuum columns have to be chosen selectively, than mixed-integer nonlinear optimization problems need to be solved. Process network flow or process synthesis problems usually fall into this category too. Examples are heat exchanger network, distillation sequencing or mass exchange networks. It should not be inferred from this list of examples that optimization is confined to large sophisticated problems occurring in large organizations. Small problems are equally important, but it may be that the role of techniques to solve problems is less important. Some small problems lack the structure required for the use of optimization techniques and for others the use of techniques would be seen as using a sledge hammer to crack a nut. A balance is required where gains to be obtained by finding optimal answers to a problem outweigh the effort required to obtain these answers.

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