Decision-making is a task that we face on a daily and continuous basis since basically all our actions come from a decision. However, in most cases we make decisions on pure instinct.
While making simple decisions by instinct or intuition is not bad, there are other decisions on important matters where it is necessary to analyze all possible alternatives.
It is important to be aware that a wrong decision can have great consequences not only personally. This is the same in the business world where a small mistake, a bad decision made, can lead us to a bottomless abyss.
From this point of view is where the importance of making reliable decisions based on facts and looking for the best decisions or in other words the optimal decisions resides.
Mathematics can provide us with many tools to support decision-making, to help us better analyze situations.
Among the models that use mathematical language, we can mention the mathematical programming models.
MODELS
Solving a problem is usually complicated, from knowing where to start to the clearest way to express the problem. The first step we must take is to discover the components, then choose those that are important and discard those that are not a fundamental part of the problem, then we must look for the relationship between these and finally select some objects or symbols that allow us to represent the simplified situation. This representation is called: model.
The model has various ways of being represented from a drawing, map, photograph, network, graph, etc. even mathematical expressions.
Among the most prominent models we have linear programming, integer programming, non-linear programming, dynamic programming and multi-objective programming.
LINEAR PROGRAMMING:
“Possibly, among the available models, the linear model is the most economically viable and the most flexible, since there is a wide variety of computer packages that allow finding solutions for a linear program. In addition, these packages are purchased at reasonable prices and do not require sophisticated computer equipment. " (Narro Ramírez, 1996)
Linear programming has been successfully tested in the chemical, agricultural, oil, automotive, forestry, metallurgical industries, in financial institutions, etc.
However, it should be mentioned that linear programming has the limitation of the linearity of the functions involved.
What is Linear Programming?
"Linear programming (PL from now on) consists of finding the values of some variables that maximize or minimize a single objective subject to a series of restrictions." (Serra de la Figuera, 2002)
PL Features:
- A single linear objective to optimize (maximize or minimize) Decision variables that are always continuous and not negative One or more linear constraints An exact knowledge of the parameters and resources used in the construction of the model
Examples of uses of linear programming:
- Optimize your food mix Optimize your chemical mix Select advertising media Select appropriate distribution channels Minimize waste management costs Helping you determine the best budget available
The general form of the linear programming model is:
?
??? Σ (? ? ? ?)
? = 1
Subject to:
?
? (?) =? 0 + Σ (? ?? ? ? ≤? ? ????? = 1….?)? = 1
? ᵢ ≥ 0
We want to find the values of the n variables x, for i from 1 to n, that allow the function called objective: c, x, +… + cn xn (represented above in summary form) to reach the maximum value, respecting the inequalities a¡, x, +… + ain xn less than or equal to the resource b¡, where each i, from 1 to m, refers to the constraints, where x¡ greater than or equal to zero establishes that the variables cannot take negative values.
The program can also be a minimization program and the inequalities or restrictions can be:>> = (greater, greater or equal, or equal to).
WHOLE PROGRAMMING:
The set of techniques available to find the best possible integer solution to a linear programming problem is called integer programming. The only difference between a linear model and an integer linear model is the restriction that some or all of the variables must be integers. To find the solution of an entire program it is necessary to use a search process in which each step must apply the process of solving a linear problem.
“An integer linear programming problem is a linear programming problem with the additional restriction that some of the variables must take integer values. When all the variables must take integer values we say that it is a pure integer linear programming problem, otherwise we say that it is mixed. We will say that a variable is binary if it can only take the values 0 and 1. A great variety of combinatorial problems can be posed as integer linear programming problems. " (UNIVERSITY OF BUENOS AIRES, 2011)
The general form of an integer linear programming model is:
?
??? Σ (? ? ? ?)
? = 1
Subject to:
?
Σ (? ?? ? ? ≤? ? To? J = 1…)
? = 1
With ? ? integer for: r ≤ i ≤ s, ? ? ≥ 0, for i = 1… n
??, ?? + 1… ??
The interpretation of this integer linear program is the same as that of the linear program, only that the values of the variables xr, xr + 1 xs must be integers.
PROGRAMMING
Sometimes and under certain circumstances it is necessary to resort to non-linear models. “There are many types of NLP problems, depending on the characteristics of these functions, so various algorithms are used to solve the different types. For certain cases where functions have simple forms, problems can be solved relatively efficiently. In some other cases, even solving small problems is a real challenge. " (Merino Master)
The general nonlinear programming model is expressed:
Max f (x)
Subject to:
gᵢ, (x) ≤ bᵢ and hᵢ (x) = 0, for: 0 ≤ i ≤ m
x ≥ 0 gᵢ, (x) ≤ bᵢ hᵢ (x)
Where f (x) (objective function) wants to be maximized respecting the relations g¡ (x) <b¡ (inequality restrictions) and h¡ (x) = 0 (equality restrictions), for values not
The objective function can be minimized instead of maximized, and the inequality constraints can be> (greater than or equal to).
When the integrity constraint is added for some of the variables the model is called an integer nonlinear program.
Nonlinear programming does not have an algorithm that solves all the problems that conform to this format, but there are packages such as Gino and Gams that have been used with good results to solve these types of problems. These packages lead to an approximate solution, that is, close to optimal.
PROGRAMMING BY OBJECTIVES
This model is based on establishing a numerical goal for each of the objectives to be achieved, formulating a relationship that represents each objective and seeking a solution that minimizes the difference between the value of each objective function expressed as a relationship between the variables and the goal to be achieved.
This model has two types of restrictions; target constraints and resource constraints.
"In general, the LP problem consists in finding the maximum or minimum of a certain (linear) function subject to a series of restrictions (all of them linear) in the form of equations or inequalities." (VILLALBA, 1974)
Similarly, there are two types of models with this structure:
- The first is called a program by objectives without priorities in which all the objectives are equally important The second gives different importance to each objective and to reflect these priorities a different weight is assigned to each deviation that appears in the objective function of the program correspondent.
The general form of the objective linear programming model is:
??? Σ (? ? ? ? +? ? ? ?)
? = 1 Subject to:
?
Σ (? ?? ? ? ≤? ?, ????: = 1,…,?)
? = 1
?
Σ (? ?? ? ? +? ?? ? ≤? ?, ????: = 1,…, ?????? ?? ?????????)
? = 1
? ? ,? ? ,? ? ≥ 0, ????:? = 1,…,?,? = 1,…, ?????? ?? ?????????
where p¡ is the weight assigned to the amount f¡ that remains to reach the goal M¡; q¡ is the weight assigned to the amount S¡ that is left over from the goal M¡. You want to minimize the sum of deviations with weights that reflect their importance. Mk is the value assigned to target k.
DYNAMIC PROGRAMMING
“Dynamic programming not only makes sense to apply it for efficiency reasons, but also because it presents a method capable of efficiently solving problems whose solution has been addressed by other techniques and has failed. Where dynamic programming has the greatest application is in solving optimization problems. In this type of problem, different solutions can be presented, each one with a value, and what you want is to find the optimal value solution. " (MALAGA UNIVERSITY)
This type of programming breaks the original problem down into simpler problems that can be solved by taking a single decision on each one. A dynamic programming problem has the following characteristics:
- It can be divided into stages and each of them corresponds to a decision making. Each stage has a finite number of possible conditions in which the system can be found. The decision policy has an effect on the transformation of the current state into a state. associated with the next stage. The solution procedure is designed to find the optimal solution for the entire problem. Knowledge of the current state of the system expresses all the information of its previous behavior and this information is necessary to determine the optimal policy thereafter. The solution procedure begins by finding the optimal solution of the last stage since this includes the optimal decision of each stage There is a recursive relationship that identifies the optimal policy for stage n,given the optimal policy for stage n + 1. When using this recursive relationship, the solution procedure moves backward stage by stage, each time finding the optimal solution for that stage, until the optimal solution is found from stage initial.
The general form of the dynamic programming model is:
?? (??) = ???
Where ? ? (? ?) It is the step ka min last
Starting from the state? ? and ? ? +1 (?? +1) is the min from Stage k + 1 to the last starting from the state? ? +1
- ADVANTAGES OF THE USE OF MATHEMATICAL MODELS:
- Some developed countries have generalized the use of these models It allows the use of mathematical instruments already developed in the achievement of a solution It can minimize costs without neglecting quality It provides a systematic, explicit and efficient way to find a solution It allows to evaluate different feasible solutions It helps to make the best decision.You can predict and compare the behavior of the current situation against various alternatives.
CONCLUSION :
Today in the industry you need security regarding the decisions to be taken; since making a bad decision involves significant costs and consequences in most cases.
However, not only on an industrial level but on a personal level we are always looking to make the optimal decision, that is, the best one for all the problems that arise. However, many times we allow ourselves to be guided by our intuition and this does not always represent a benefit.
Today there are various mathematical models available accompanied by computational support that allow us to make optimal decisions with relative ease. It is worth mentioning that these mathematical models with computational support can perform all possible combinations of strategies and determine the best of them. Thus, they give us the optimal solution to the problem and based on it we can make the best decision.
THANKS:
I thank my alma mater the Technological Institute of Orizaba, Professor Fernando Aguirre y Hernández who teaches the subject of Fundamentals of Administrative Engineering for showing us that we are capable of writing articles on various topics, for encouraging the habit of reading and above all for helping us realize what we are capable of achieving.
BIBLIOGRAPHY
- Merino Maestre, M. (sf). CLASSIC OPTIMIZATION TECHNIQUES. UPV / EHU.Narro Ramírez, AE (1996). Application of some mathematical models to decision making. Politics and Culture, 183-198, Serra de la Figuera, D. (2002). Quantitative Methods for Decision Making. Banco Bilbao Vizcaya Foundation and CRES.UNIVERSIDAD DE BUENOS AIRES. (2011). Combinatorial Optimization. Obtained from http://cms.dm.uba.ar/UNIVERSIDAD DE MALAGA. (sf). LANGUAGES AND COMPUTER SCIENCES. MALAGA UNIVERSITY. Obtained from PROGRAMACIÓN DINAMICA: http://www.lcc.uma.es/~av/Libro/CAP5.pdfVILLALBA, D. (1974). PROGRAMMING BY OBJECTIVES. SPANISH JOURNAL OF FINANCING AND ACCOUNTING, 369-388.
