The Problem
I need to make it more durable. I need to make it lighter. I need to lower the cost of materials.
I need to do all three at the same time. Is there a way?
Traditionally, there have been problems that are so complex as to be imponderable because of the large number of variables and constraints involved. Many of these can now be solved by combining accurate, sophisticated models developed from real data with optimization programs to give solutions that provide the means for operating your business at or near optimal performance.
The Solution
Nonlinear Synergetics offers a number of critical tools for full nonlinear optimization modelling. For the case of a single objective function, for either minimization or maximization requirements, we can use up to 1000 design variables with up to 1000 full nonlinear constraints. The constraints can be any mixture of inequality and equality constraints. The Nonlinear Synergetics vector step pattern search methodology allows such complexity without a significant time penalty, and proves or disproves solutions to problems with either single or multiple objectives.
Exterior Vs. Interior. Most problem solvers begin with an algorithm comprised of preconceived notions - a belief that the eventual solution must reside within previously-determined parameters. Our approach begins within a reasonable range "outside the box", allowing for results that can be both innovative and unexpected. Nothing remains untested.
With the NS approach, constraints are not met until the end of the process. The upshot is that the objective function in the case of minimization moves from a less efficient, non-optimal value and rises up to “kiss” the optimal solution region. For maximization, the values float downward onto the region. Because these features are part and parcel of the basic algorithm, most solution cases result in the Global or the best solution possible to the problem, within the boundaries of the constraints and the design variables.
Identifying the Pareto Variables. Any complex optimization problem contains critical, key variables that sit right alongside the remaining variables whose affect on the optimal result is inconsequential. Therefore a seemingly large problem, believed to be the result of many variables, is reduced to a simpler, more compact and meaningful representation of only the essentials needed to model the problem. These key elements are what NS calls the "Pareto Design Variables". Identifying the critical, Pareto variables early is key to finding workable solutions within a reasonable period of computational cycles. Further, the Pareto variables are ranked according to their influence, thus increasing the likelihood of rapidly finding a global or near-global solution.
Meeting Multiple Objectives
Additionally and most importantly, Nonlinear Synergetics offers the pinnacle of Nonlinear Programming - that of meeting the requirements of multiple objectives. Many situations have two or more antagonistic goals that need to be achieved. Let's take a closer look at this.
A real-life example. A given manufacturing plant must produce maximum profit and maximum market share accordingly to the view of management and the company's shareholders. Alternatively, the employees' union would like to see maximum salaries and maximum numbers of union members employed. Clearly the objectives of management and the aspirations of the union are at odds. This common business example clearly has four objective functions involved: profit, market share, maximum salaries, and maximum union membership.
Nonlinear Synergetics can solve this most complicated problem given that the model can be developed and verified and agreed to by management and the union. The solution would represent a compromise and would predict the profit level, the market share, the salary range and the likely union membership employed. Also, because Nonlinear Synergetics employs a reference point method, the minimum requirements of profit, market share, salary and staffing can be located as a minimal requirement starting point that must be met initially for both Management and the Union. The reference points also serve as a benchmark limit for each individual objective allowing the further complication of scaling or weighting of objective functions if needed.
Clearly our example here demonstrates the strong application of nonlinear programming to the most demanding real problem scenarios of the world at large. There are thousands of such examples within the realms of engineering, business, agriculture, etc. – virtually an endless landscape.
There is a way to solve complex problems with multiple objectives. To find out how, contact Nonlinear Synergetics today.