Production Planning
Reality as a Mathematical Optimization Model
It is one of the oldest subfields of artificial intelligence: mathematical optimization. Why it’s essential to modern production planning.
Mathematical optimization, as a subfield of artificial intelligence, is the best suited method to solve complex real-world problems. Developed within the scientific field of operations research, it is the core of modern optimization software. It is used to transfer the reality of complex processes into a mathematical model. From planning production sequences and transport sequences to procurement and sales planning processes, this mathematical model describes the entire range of all possible decisions. A corresponding algorithm then looks at these to find the best decision for a given objective, taking into account all restrictions.
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Decision variables are things to adjust! Constraints are limitations! Mathematical optimization is about: Make the best out of the limitations!
The right answer to the crucial question
Imagine you are a production planner. Your goals are strict adherence to delivery deadlines, short lead times, and low costs. In practice, however, tight delivery deadlines, short product life cycles, and numerous product versions lead to complex production. It is almost impossible to keep track of the interdependencies of individual production steps for preliminary, intermediate, and end products and the resulting restrictions in terms of personnel, material, and machine capacities. There are, therefore, an unmanageable number of answers to the crucial question of when which production orders should be processed in which sequence and on which machines. Mathematical optimization is the only way to find the one answer that achieves the best results in terms of adherence to schedules, throughput times, and costs. But how does it work? What goes on behind the scenes of specialized production planning software?
Three steps
Mathematical optimization has three steps: the formulation of a real-world problem as a mathematical model, the development of algorithms to solve these mathematical models, and the development of software programs to run and create the models and algorithms For example, the mathematical model of a production planning problem includes decision variables and fixed parameters as well as all the relationships that exist between them. Decision variables basically represent the questions that every production planner must constantly ask, such as:
- When and on which machine should an operation be scheduled?
- Can I use a component or not?
- Should the maintenance of a machine take place today or tomorrow?
- When should a customer be supplied?
These and many other questions can be formalized in a mathematically descriptive way. Each answer to one of these questions is a decision, but one in which there is leeway. Therefore, they are the variables in the mathematical model. The leeway gained from having variables is, however, restricted by fixed basic conditions, such as personnel and machine capacities, set assembly sequences, material availability, and much more. These deterministic parameters now correlate to the decision variables. Whether a component can be used or not depends, among other things, on whether it is already available. Such dependencies can be mathematically formulated relatively easily in the form of what are known as "restrictions." In addition, the model is given an objective function that must be minimized or maximized. Depending on the strategy, companies ask themselves which production plan is the best for minimizing production costs or maximizing on-time delivery.
The core function of mathematics
Mathematics provides the answer needed. Ultimately, it is a matter of determining the extreme values of a function, which is one of mathematics’s core functions. Determining extreme values considers the numerous constraints, which are looked at in the form of equations or inequalities. In mathematical terms, the core of optimization, therefore, lies in solving extreme value problems in the form of a mostly linear function with equations or inequalities as constraints. This is also known as "linear programming."
A decision with a view of the entire system
The mathematical model does not just map the production process. In fact, it outlines a solution space for an optimization problem that can be examined using an algorithm. Simply put, the algorithm performs a continuous mathematical proof that shows that the best solution must lie in range A and not in range B of the solution space. In this way, the algorithm gradually moves closer to the best possible production plan. As a result, good algorithms always keep the entire system in mind when proposing these decisions. That is why when it comes to questions about machine utilization, it is not necessarily best to schedule the next follow-up order on a free machine. Instead, an algorithm may recognize that it could be better to let the next job wait another five minutes, as this will free up a machine that is more suitable for the job. During this time, the relevant software runs automatically in the background. Production planners do not have to be math geniuses. Rather, they are able to organize the production processes based on a broad base of knowledge. Wherever necessary, they can override the software's suggestions at any time.
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Nobody has to be a math genius to work with mathematical optimization software!
The importance of mathematical optimization is growing
Mathematical optimization will play an even more important role in the future. Especially because when used with other AI technologies, its effectiveness will continue to grow. Machine learning, in particular, should be mentioned here. It is used to determine relevant data and thus gain better input for the creation and calculation of models. In parallel, algorithmic research will also continue to accelerate algorithms so that optimal plans for increasingly complex production processes can be calculated at the required speed in the future.
Operations Research: In the Slipstream of AI
Mathematical optimization is part of operations research. At INFORM, operations research was and is the basis for the development of specialized optimization software.
Read here how INFORM was one of the first companies to make operations research useful for economic applications in regard to software systems.