A brief history of OR

Operations Research originated in the efforts of military planners during World War II. In the decades after the war, the techniques began to be applied more widely to problems in business, industry and society.

Terminology

Mathematical modeling

Feasible solution(s)

Optimization

Iterative computations

Attack the problem with the following three questions

1. What are the decision alternatives?
2. Under what restrictions is the decision made?
3. What is an appropriate objective criterion for evaluating the alternatives?

Before we dive in, lets take an example

You have pieces of wire of length L inches and you want to form a maximum-area rectangle out of the pieces, what shroud be the width and height of the rectangle ?

In this problem, the alternatives are the width and height of the rectangle, so lets define them:

w = width of the rectangle in inches
h = height of the rectangle in inches

Now, expressing the restrictions verbally:

1. Width of rectangle + Height of rectangle = Half the length of the wire
2. Width and height cannot be negative

This translates to:

1. 2(w+h) = L
2. w, h >= 0

and lastly, the objective of the problem is the maximization of the area of the rectangle. Let z be the area of the rectangle, then the complete model becomes:

Maximize z = wh

Subject to:

2(w + h) = L
w, h >= 0

The optimal solution for this problem is w = h = L/4, which means it's a square.

Now, based on the preceding problem, we can generalize any OR model like this:

A solution of the model is feasible if it satisfies all the constraints. It is optimal if, in addition to being feasible, it yields the best (max or min) value of the objective function

Solving of the OR model

In OR, there is no single general technique to solve all mathematical models that can arise in life. Instead, the type and complexity of the mathematical model dictates the nature of the solution method.

The most prominent OR technique is Linear Programming. It is designed for models with linear objective and constraint functions. Other techniques include Integer Programming (in which the variables assume integer values), Dynamic Programming, Network Programming, Nonlinear Programming.

In most OR techniques, solutions are not obtained in closed forms. Instead, they are determined by algorithms. An algorithm provides fixed computational rules that are applied repetitively to the problem, with each repetition (called iteration) moving the solution closer to the optimum. Now, since these computations are tedious, they're executed on the computer most of the time, however, it is important to do them by hand a few times in order to understand how the algorithms are implemented.

Some mathematical models may be so complex that it is impossible to solve them by any of the available optimization alogorithms. In such cases, it may be necessary to abandon the search for the optimal solution and simply seek a good solution using heuristics or rules of thumb. A good solution is not an optimal solution, but it is very close to it.

Phases of an OR study

Definition of the problem

Construction of the model

Solution of the model

Validation of the model

Implementation of the solution

Problem definition involves defining the scope of the problem under investigation. This function should be carried out by the entire OR team. The aim is to identify three principal elements of the decision problem: (1) description of the decision alternatives, (2) determination of the objective of the study, and (3) specification of the limitations under which the modeled system operates.

Model construction entails an attempt to translate the problem definition into mathematical relationships. If the resulting model fits one of the standard mathematical models, such as linear programming, we can usually reach a solution by using available algorithms. Alternatively, if the mathematical relationships are too complex to allow the determination of an analytic solution, the OR team may opt to simplify the model and use a heuristic approach. or they may consider the use of simulation, if appropriate, In some cases, mathematical. simulation, and heuristic models may be combined to solve the decision problem.

Model solution is by far the simplest of all OR phases because it entails the use of well-defined optimization algorithms. An important aspect of the model solution phase is sensitivity analysis. It deals with obtaining additional information about the behavior of the optimum solution when the model undergoes some parameter changes. Sensitivity analysis is particularly needed when the parameters of the model cannot be estimated accurately. In these cases, it is important to study the behavior of the optimum solution in the neighborhood of the estimated parameters.

Model validity checks whether or not the proposed model does what it purports to do, that is, does it predict adequately the behavior of the system under study? Initially, the 0R team should be convinced that the model’s output does not include “surprises.” In other words, does the solution make sense? Are the results intuitively acceptable? On the formal side, a common method for checking the validity of a model is to compare its output with historical output data. The model is valid if, under similar input conditions. it reasonably duplicates past performance. Generally, however, there is no assurance that future performance will continue to duplicate past behavior. Also, because the model is usually based on careful examination of past data, the proposed comparison is usually favorable. If the proposed model rep resents a new (nonexisting) system, no historical data would be available. In such cases, we may use simulation as an independent tool for verifying the output of the mathematical model.

Implementation of the solution of a validated model involves the translation of the results into understandable operating instructions to be issued to the people who will administer the recommended system. The burden of this task lies primarily with the OR team.

That would be it for now . In the next post I will jump into Modeling with Linear programming.

Thanks for reading!