Mathematical model

When one have to solve a decision making problem, the first (and most important) step is to analyze the problem to build a mathematical model. This process of reflection is called « scientific method » which consists of five steps:

  1. Mathematical aspects: the specifications of the analysis highlight the constraints and objectives. The constraints and objectives must be written using algebra and logic.
  2. Modeling: following the mathematical understanding of the problem, the modeling step consists in finding the best tool to represent the model. These tools include graph theory, game theory, linear programming, constraint programming, and so on.
  3. Analysis and resolution: following modeling, one must be able to analyze the complexity in time and space of the model after having modeled it from an algorithmic point of view.
  4. Implementation and results: we must find the best way to implement the algorithm in programming method or language to create the most relevant and effective software possible.
  5. Deployment solutions: the last step is to validate the model created by the resolution of other problems in the same form as the original.

The mathematical model is therefore the basis of all engineering work. A mathematical model is characterized by three notions:

  1. The presence of choices, to be made among a finite set (a notion that can be compared to the theory of probability)
  2. Some constraints defining the finite set of choices
  3. A system to evaluate the choices

Example 1

An businessman has three manufactures and two products. Each batch of product earns him an amount of money, and he knows the number of hours for the manufacture of each type of batch in his factories.

decision1

The businessman wants to maximize his profit, so we must find the best possible production.

Let us set decision variables:

  • x1 = the number of batch of product 1
  • x2 = the number of batch of product 2

Let us set the objective function:

  • z = the total profit (in thousands of euros)
  • z = 3 x1 + 5 x2 (from the last raw of the table)
  • max z, we are looking for the maximum value that z

Let us set the constraints:

  • x1 ≤ 4 (second raw of the table)
  • 2 x2 ≤ 12 (third raw of the table)
  • 3 x1 + 2 x_2 ≤ 18 (fourth raw of the table)
  • x1 ≥ 0 and x2 ≥ 0 (the number of batches is positive or null)

Which gives the following mathematical model:

decision2

What can be represented from a graphical point of view (the choice space is in gray):

decision3

Example 2

Now that the businessman knows how to optimize his profit, he tries to minimize his expenses. These are composed to employee wages and working hours. The manufacturer estimates the minimum number of employees (MinEmp) to be assigned during each period of the day. Each employee must work shifts in order to maximize his or her time of attendance, one day has four quarter (Quart), and these quarters require a remuneration. The data set is described in the following table:

decision4

Let us set decision variables:

  • x1= the number of employees on the first quarter
  • x2= the number of employees on the second quarter
  • x3= the number of employees on the third quarter
  • x4= the number of employees on the fourth quarter
  • x5= the number of employees on the fifth quarter

Let us put the objective function:

  • Z = the total cost
  • Z = 170 x+ 160 x+ 175 x3+ 180 x4+ 195 x(from the last raw of the table)
  • min Z, we are looking for the minimum value Z can take

Let’s set the constraints:

  • x≥ 48 (from the second raw of the table)
  • xx79 (from the third raw  of the table)
  • etc.

Which gives the following mathematical model:

decision5

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