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Learn Mathematical Modeling

Mathematical modeling is the art and skill of translating real problems into mathematics. It captures the essence, the pattern and the structure of a problem and formulates them in a formal way.

For whom: Modeling language designers, Students in business analytics, operation research, operations management, and all persons interested in mathematical applications.

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3 math models

Step 01: Learn What modeling is

Learn the defintions of "model". Get familiar with other modeling concepts. Find out what a mathematical model is.

Illustration of Bohr's atom model

model ≡ representation of knowledge

A model is a description or a representation of an object, a system or a process in order to understand or explain how it works. It structures knowledge in order to make decisions, it helps to control the environment, and to predict events.

Bohr's atom model, consisting of a dense nucleus surrounded by orbiting electrons is an approximation of the "atom". The orbits are labeled by an integer, the quantum number. Electrons can jump from one orbit to another by emitting or absorbing energy. It explains how electrons can have stable orbits around the nucleus and why atoms had characteristic emission spectra. It is still used to introduce students to quantum mechanics.

Illustration of an abstraction

modeling ≡ problem solving ≡ science

Modeling means capturing the essence, the pattern and the structure of a problem and to formulate it in a precise way. It makes abstraction of the unessential.

Planning to install semaphores on a street junction is a problem that can be translated into vertex coloring of a graph: The street lanes are mapped to the vertices and conflicting lanes to edges. Vehicules on lanes with the same color can traverse the junction at the same time without collision. The goal is to have the least number of colors (semaphore phases).

People with modeling skills are in high demand, because formal models are pervading the industrial and economical processes: planning workforce, production planning, route planning, sport scheduling.

Illustration of mathematical models

Math modeling ≡ Insight, not numbers

The power of abstraction, this ability to leave out some or many details about a concrete problem in order to use the "distillate" for a multitude of similar problems, is one of the most striking characteristics of our mind. Mathematics and logic are essential tools in the process of abstraction.

A mathematical model consists of symbols like operations, numerical data, variables (the unknowns), and constraints. It is the results of a abstraction process and is a powerful language to express, represent, analyse, and provide insight into real-world phenomena. Once a real problem is formulated as a mathematical model, it can be solved using powerful computer tools.

Step 02: Learn how to build models

Building models is a skill that must be learnt. A lot of experience is needed to create good models. The huge material proposed here gives useful guidelines by means of many case studies. Five simple examples are presented below.

01 very easy

How old is Peter ?

Peter's age is a third of his father's age, but in 10 years his father will only be two times as old as Peter will be. What is Peter's age today?

(This is the kind of problems the reader probably has seen in primary school in mathematics. Is it easy? Then go to the next, otherwise follow carefully the explanation in the "solution" link below!)

Illustration: How old is Peter problem

02 easy

3-jugs problem

There are 3 jugs with capacities of 8, 5, and 3 liters. Initially, the 8-liter jug is full of water, whereas the others are empty. Find a sequence for pouring the water from one jug to another such that the end result is to have 4 liters in the 8-liter jug and the other 4 liters in the 5-liter jug. When pouring the water from a jug A into another jug B, either jug A must be emptied or B must be filled.

(This problem is a gentle introduction of graphs, an important concept in modeling in general.)

Illustration: 3-jugs problem

03 easy, but...

Seven digits puzzle

Use each digit from 1 to 7 exactly once, and place them into the circles A to G of the Figure in such a way that the sum along each of the five straight lines is the same.

(One may solve this problem without mathematics just by trying. The mathematics here is a little bit more involving. The concept of indexes is used. Not familiar with them? Then go to Help. More help is given there.)

Illustration: 7-digits problem

04 not so easy

The clock problem

Divide the clock with a straight cut into two parts such that the sum of the numbers in both parts are equal?

(This problem seems particularly easy. It is not! Well by trying the reader get quickly a solution. But then the question arises: It this te unique solution? How can one find all? The resulting mathematical model uses binary variables and logical conditions. Doesn't sound familiar? Then go to Help. More help is given there.)

Illustration: The clock problem

05 difficult

Six snakes problem

The 6-snakes problem is a board game for one person. The board is a hexagon with 72 notches where the 72 pellets must be placed. The pellets are divided into 6 groups of 12 pellets with the same color. In each group, exactly one pellet is a little bit larger that is called the "head of the snake". The inital situation of the game is to place the 6 heads of the snakes anywhere in a notch of the board. The goal is to place all other pellets such that the pellets with the same color form a "snake" on the board starting with the head of the snake, that is, each following pellets with the same color must be neighbor of the previous pellet with the same color and the snake cannot cross over another snake.

Illustration:6-snakes problem

Step 03: Learn how to implement models

Quickly a model becomes too complicated to be solved manually. The problem then is to find the right tools to formulate the problem on a computer. Basically, what we need is:
→ a software that represents the problem and
→ a software that solves the model.

The main tool used here to represent a model is the modeling language called LPL. The basics are easy to learn and initially there is no need to install any software. The models can directly be solved through the Internet. (Other tools are presented later.)

How old is Peter (Solution)

The complete mathematical model is as follows:

It is easy to solve these two equations by hand: (1) Substitute y in the second equation by 3x, and then (2) solve it for x.

LPL code for the problem

  model AgeOfPeter "How old is Peter";
    variable x; y;
    constraint A: y = 3*x;
               B: y+10 = 2*(x+10);
    Write('The age of Peter is: %d\n', x);

Copy this code and paste it HERE.
Then click the button "RUN and SOLVE".

Seven digits puzzle (Solution) ?

The complete mathematical model is as follows:

It is a little bit more complicated to solve this probem.

LPL code for the problem

  model Puzzle7 "The 7-Digits Puzzle";
    set i,j := ['A','B','C','D','E','F','G'];
    alldiff variable x{i} [1..7]; variable z;
      L1: x['A'] + x['D'] + x['G'] = z;
      L2: x['A'] + x['C'] + x['F'] = z;
      L3: x['B'] + x['E'] + x['G'] = z;
      L4: x['B'] + x['D'] + x['F'] = z;
      L5: x['C'] + x['D'] + x['E'] = z; 
    Write('%s , z=%d\n',
      {i} Format('%s=%d ', i, x), z));

Copy this code and paste it HERE.
Then click the button "RUN and SOLVE".

The clock problem (Solution) ?

The complete mathematical model is as follows:

There is already some modeling knowledge needed to understand this formulation.

LPL code for the problem

  model DivideClock;
    set i := 1..12;
    binary variable x{i};
    constraint A: sum{i} i*x = (sum{i} i)/2;
         B: exactly(2){i} (x[i]<>x[i%#i+1]);
    Write('Numbers: %3d on one side\n',
       {i|x} i);

Copy this code and paste it HERE.
Then click the button "RUN and SOLVE".

Step 04: Further Readings

Indexing is one of the most fundamental concept in mathematical notation. It is extremely powerful and allows the modeler to concisely formulate large and complex mathematical model. In this paper, the reader learns it from the ground up.

On Indexing →

Models can be classified into groups using various criteria: discrete, continuous, linear, non-linear, etc. Diverse models type also dominate different research domains. This paper gives an overview from the perspective of operations research.

Classifying models →

Logical and Boolean conditions are often used in real-live problem solving. However, its importance in practical modeling contrasts sharply with its treatment in modeling books and courses. Many modeling techniques and applications are exposed in this paper.

Logical modeling →

Pivot tables are a powerful means to represent multi-dimensional data on a two-dimensional space. They enable a user to show a large amount of structured data from different angles and perspectives.

Pivoting Tables →

This text give a (subjective) overview of mathematical modeling tools and software and implements for each system a small model to give a flavor of its syntax. Each model is also implemented in LPL to compare without giving a rating.

Modeling tools →

My Vision of a modeling language is exposed here. I consider my contribution -- namely LPL -- as a (small) subset of that fully-fledged language. The paper does not present a formal statement, it is rather a compilation of some ideas

(The paper is not yet ready.)

My Vision →