Into the Unknown
When you are given a linear program, in many cases we call it a primal linear program. It is usually going to be able to find another program, called a dual linear program. The primary reason why we want to use duality is that the computation time of the simplex method is roughly proportional to…
When it comes to Operations Research, we are mainly talking about optimization problems, so the theories are mainly optimality conditions. They basically mean for an optimization problem, somehow there should be an optimal solution. We want to get the conditions which will make the optimal solution satisfy. Typically we search for the necessary conditions, while…
Operations Research (2): Optimization AlgorithmsNational Taiwan University You probably have used some solver / optimizer software to solve problems, this wonderful course will show you what is under the hood. Being able to comprehend what has happened is crucial to customize the algorithms or even to develop your own ones. There is a warm-up at…
The objectives for any research is that we want to build a mathematical model to formulate the given problem. The heart of operations research is a mathematical model. That model is going to precisely describe the problem, but the model may be too complicated to get an optimal solution. We need heuristic algorithms, and the model could be…
Non-Linear Programs When visualizing a linear program, its feasible region looks like a polygon. Because the objective function is also linear, the optimal solution is on the boundary or corner of the region. But a non-linear program is quite different, its feasible region may be in any shape, moreover the optimal solution may not exist…
In some cases, variables must take integer values, or binary values. Formulating and solving the models with integer variables is integer programming. There are mainly 2 methods to solve it: The branch-and-bound algorithm finds an optimal solution for an integer program. Essentially, it decomposes the integer program into multiple linear programs, then solves all of…
The simplex method is the most fundamental tool in linear programming, it is a single algorithm that is able to solve any kinds of linear program regardless of its format, number of variables and constraints. We firstly transform the linear program to a standard form, then we focus on the standard form. The idea of…
There are 2 perspectives to look at linear equation systems, row view and column view. Both are equivalent, sometimes row view is more intuitive, sometimes column view is easier to understand. Linear Equations (n = 2) By row Each equation represents a straight line on the x-y plane.The intersection is the only point on both…
Operations Research (1): Models and ApplicationsNational Taiwan University This is an amazing beginner-level course about Operations Research (abbr. OR). Simply put, it is all about optimization and decision-making. It is deep-rooted in math, but its emphasis on practical applications makes OR even more beautiful. Firstly, the course introduces basic concepts, then 3 fundamental models are…
In many cases, we need to deal with nonlinear situations, e.g.: product pricing decision, inventory, portfolio optimization. Above all, the purpose is to find a balance between tradeoffs. Usually non-linear programs are more general than the linear programs. Formulating non-linear program is usually easy because you rarely use weird constraints; but its optimization would be…
Integer programming is generally from linear programming but allowing you to have integer variables, which means you may restrict the values of your variables to be integers. There are some problems linear programming is not suitable but integer programming can be helpful, for example: setup cost / time, facility location, machine scheduling, vehicle routing. All…
Almost all of problems that Linear Programming is able to solve is about resource allocation, for example: product mix, production, inventory, scheduling, etc. You have limited resource, and so many things to do. So that you have decision-making problems. Linear program is a special type of mathematical program with some special properties. Mathematical Programs Basically,…
Motivations Management is the attainment of organizational goals in an effective and efficient manners through planning, organizing, leading, and controlling organizational resources. Daft R. L. (2014) Management When you run a company or an organization, you want to do something, that is your goal, but you have limited resources, you want to assign or allocate…