Simulated annealing

Metaheuristics are general problem solving algorithms which abstract from the actual problem description. Therefore they can be easily applied to many optimization problems. Simulated annealing is a simple and fast metaheuristic with an analogy to metal processing. As metal particles generate a solid and regular structure when cooling slowing simulated annealing seeks a low-energy solution avoiding local optima. 1 NP-complete problems in computer science Since Stephen Cook’s paper The complexity of theorem proving procedures from 1971 computer scientists know about a special class of problems which are very difficult to solve. These problems were later called NP-complete problems. Richard Karp showed in 1972 that there are some very important and often needed problems among them, such as the 0-1 integer programming problem. Many problems in the area of business studies and operations research can be reduced to a combinatorial optimization problem and then be solved by an integer programming algorithm. Capacity planning is such a problem. Imagine a big car company which has several production sites all over the world. A car typically consists of about 10000 components. The difficulty is to decide which component to produce at which site. It might be cheap to produce components in countries which are rich in raw materials or countries having low unit labor costs but some production processes might only be feasible in industrialized nations. Furthermore the finished cars need to be transported to the countries where they can be sold. By minimizing expenses and maximizing profits good solutions can be found. Such a problem can be expressed mathematically by a set of integer inequalities. This mathematical representation, also called a constraint optimization problem, is a combinatorial optimization problem and can be solved by linear integer programming, which is a generalization of 0-1 integer programming. Thus the capacity planning problem is NP-complete. Unfortunately NP-complete problems can’t be solved efficiently so far and they will most likely never be solved efficiently. In this context efficiently denotes polynomial computation complexity.