(Introducción a la investigación de operaciones9a edición, Frederick Shillier)Las raíces de la IO pueden encontrarse muchas décadas atrás, cuando se hicieron los primeros intentos por emplear el método científico para administrar una empresa. Sin embargo, el inicio de la actividad llamada investigación de operaciones es atribuible a ciertos servicios militares que se prestaron al inicio de la Segunda Guerra Mundial. Debido a los esfuerzos bélicos, existía la urgente necesidad de asignar recursos escasos a las distintas maniobras militares y a las actividades que componían cada operación de la manera más eficaz. Por ello, las administraciones militares estadounidense y británica llamaron a un gran número de científicos para que aplicaran el método científico a éste y a otros problemas estratégicos y tácticos. En realidad, les solicitaron que hicieran investigación sobre operaciones (militares). Estos grupos de científicos fueron los primeros equipos de IO. Debido al desarrollo de métodos eficaces para utilizar la nueva herramienta que representaba el radar, los científicos contribuyeron al triunfo en la guerra aérea que libró Gran Bretaña.Sus investigaciones para mejorar el manejo de las operaciones antisubmarinas y de protección también tuvieron un papel importante en la victoria de la campaña del Atlántico Norte. Esfuerzos similares fueron de gran ayuda en la campaña del Pacífico.Al terminar la guerra, el éxito de la IO en las actividades bélicas generó gran interés debido a las posibilidades de aplicarla en un ámbito distinto al militar.

Como su nombre lo indica, el objetivo de esta disciplina implica “investigar sobre las operaciones”.En consecuencia, esta disciplina se aplica a la problemática relacionada con la conducción y la coordinación de actividades en una organización. En esencia, la naturaleza de la organización esirrelevante, por lo cual la IO ha sido aplicada de manera extensa en áreas tan diversas como manufactura, transporte, construcción, telecomunicaciones, planeación financiera, cuidado de la salud, fuerzas armadas y servicios públicos, por nombrar sólo unas cuantas. Así, la gama de aplicaciones es inusualmente amplia.La IO incluye el término investigación en el nombre porque utiliza un enfoque similar al que se aplica en las áreas científicas establecidas. El método científico se utiliza para explorar los diversos problemas que deben ser enfrentados, pero en ocasiones se usa el término management science o ciencia de la administración como sinónimo de investigación de operaciones. El proceso comienza por la observación cuidadosa y la formulación del problema, lo cual incluye la recolección de los datos pertinentes. El siguiente paso es la construcción de un modelo científico —generalmente matemático— con el cual se intenta abstraer la esencia del problema real. En esta etapa se propone la

hipótesis de que el modelo será una representación tan precisa de las características esenciales de la situación, que permitirá que las conclusiones —soluciones— que se obtengan sean válidas también para el problema real. Después se llevan a cabo los experimentos adecuados para probar esta hipótesis, para modificarla si es necesario y para verificarla en determinado momento, paso que se conoce como validación del modelo. En cierto sentido, la IO involucra la investigación científica creativa de las propiedades fundamentales de las operaciones. Sin embargo, es más que esto. La IO se ocupa también de la administración práctica de la organización. Por lo tanto, para tener éxito, también debe proporcionar conclusiones claras que el tomador de decisiones pueda usar cuando sea necesario.Otra característica de la investigación de operaciones es su amplio punto de vista. Como quedó implícito en la sección anterior, la IO adopta una visión organizacional. Desde esta perspectiva intenta resolver los conflictos de intereses entre los componentes de la organización de forma que el resultado sea el mejor para ésta en su conjunto. Ello no significa que el estudio de cada problema deba considerar en forma explícita todos los aspectos de la organización, sino que los objetivos que se persiguen deben ser congruentes con los objetivos globales.Una característica adicional de la investigación de operaciones es que intenta encontrar una mejor solución —llamada solución óptima— para el problema en cuestión. (Se dice una mejor solución y no la mejor solución porque es posible que existan muchas soluciones que puedan considerarse como las mejores.) En lugar de conformarse con mejorar el estado de las cosas, la meta es identificar el mejor curso de acción posible. Aun cuando debe interpretarse con todo cuidado en términos de las necesidades reales de la administración, esta “búsqueda del mejor camino” es un aspecto importante de la IO.Estas características conducen de manera casi natural a otra. Es evidente que no puede esperarse que un solo individuo sea experto en los múltiples aspectos del trabajo de investigación de operaciones o de los problemas que se estudian, sino que se requiere un grupo de individuos con diversos antecedentes y aptitudes. Cuando se decide emprender un estudio de IO completo de un problema nuevo, es necesario emplear el enfoque de equipo. Este grupo de expertos debe incluirindividuos con antecedentes sólidos en matemáticas, estadística y teoría de probabilidades, al igual que en economía, administración de empresas, ciencias de la computación, ingeniería, ciencias físicas, ciencias del comportamiento y, por supuesto, en las técnicas especiales de IO. El equipo también necesita experiencia y aptitudes necesarias para considerar de manera adecuada todas las ramificaciones del problema dentro de la organización.

Fue hasta 1947 cuando la programación lineal unificó todos los diversos problemas estadísticos y económicos, entre otros campos, proveyendo un marco de referencia matemático y un método computacional, el método simplex, para formular tales problemas y determinar las soluciones eficientemente. Este desarrollo coincide con la creación de las computadoras digitales, que muy rápidamente se convirtieron en herramientas necesarias en la aplicación de la programación lineal en áreas donde sin el uso computacional no hubieran sido factibles.

The diet problem was one of the first optimization problems studied in the 1930s and 1940s. The problem was motivated by the Army's desire to minimize the cost of feeding GIs in the field while still providing a healthy diet. One of the early researchers to study the problem was George Stigler, who made an educated guess of an optimal solution using a heuristic method. His guess for the cost of an optimal diet was $39.93 per year (1939 prices). In the fall of 1947, Jack Laderman of the Mathematical Tables Project of the National Bureau of Standards used the newly developed simplex method to solve Stigler's model. As the first "large scale" computation in optimization, the linear program consisted of nine equations in 77 unknowns. It took nine clerks using hand-operated desk calculators 120 man days to solve for the optimal solution of $39.69. Stigler's guess was off by only $0.24 per year

George B. DantzigInterfacesVol. 20, No. 4, The Practice of Mathematical Programming (Jul. - Aug., 1990), pp. 43-47

The following example is a typical programming problem which can be formulated linearly; it is well to have them in mind before reaching out other concepts.(Chapter 1 page 4 George B. Dantzig, Linear Programming and extensionsThe housewife’s problem.

A family of five lives on the modest salary of the head of the household. A constant problema is to determine the weekly men after due consideration of the needs and tastes of the family and the price of foods. The husband must have 3,000 calories per day, the wife is on a 1,500 – calorie reducing diet, and the children require 3,000 2,700 and

2,500 calories per day, respectively. According to the prescription of the family doctor, these calories must be obtained for each member by eating no more than a certain amount of fats and carbohydrates and not less than a certain amount of proteins. The diet, in fact, places emphasis on proteins. In addition, each member of the househould must satisfy his or her daily vitamin needs. The problem is to assemble menus, one for each week, that will minimize cost according to Thursday food prices.

This is a typical linear programming problem: the possible activities are the purchasing of food and different types; the program is the amounts of different foods to be purchased; the constraints on the problem are the calorie and vitamin requirement of the household must satisfy and the upper or lower limits set by the physician on the amounts of carbohydrates, proteins, and fats to be consumed by each person. The number of food combinations which satisfy these constraints is very large. However, some of these feasible programs have higher costs than others. “The problem is to find a combination that minimizes de total expense” (Stigler, 1945)

(Chapter 27 page 551 dantzig) STIGLER’S NUTRITION MODEL: AN EXAMPLE OF FORMULATION AND SOLUTION.

On the first applications of the simplex algorithm was to the determination of an adequate diet that was of least cost. In the fall of 1947, Jack Laderman of the Mathematical Tables Project of the National Bureau of Standards undertook, as a test of the newly propose simplex method, the first large-scale computation in this field. It was a system with nine equations in seventy-seven unknowns. Using hand-operated desk calculators, approximately 120mand-days were required to obtain a solution.

The particular problem solved was one which had been studied earlier bye G. J. Stigler, who had proposed a solution based on the substitutions of certain foods bye other which gave a more nutrition per dollar. He then examined a handful of the possible 510 ways to combine the selected foods. He did not claim the solution to be the cheapest but gave good reasons for believing that the annual cost could not be reduced any more than a few dollars. Indeed, we shall see that Stigler’s solution, when converted from a cost-per-day to a cost-per-year, was only 24 cent s higher than the true minimum for the year, which was $3969 (10.9 per day)

PROBLEMS IN FORMULATING A MODEL

Before launching into the mathematical characteristics of the nutrition problem, it is worthwhile to see just how to develop a “mathematical model”. It will be seen to be far from a precise operation, and it is only natural to question the validity of refined techniques for solving what is clearly and approximate model. This situation is typical almost everywhere programming techniques are applied. One should remember, however, that one reason why only the approximate models exist today that has been the historic inability of the investigator to solve any large-scale complex model. As the tools for handling these systems increase, so does the desire of the investigator increase to refine his models to take advantage of the these new techniques. The next few years will probably see the end of this vicious circle of the past, in which poor model building justified rapid rough “solutions” and, conversely, the non-existence of methods of accurate solution justified poor model building. It is likely that both model building and solution techniques will begin to reinforce each them in a positive manner.

Stigler’s paper provides a very frank discussion of the background of the nutrition model; the greater part of what follows is based upon this source and follows it very closely. Stigler began with some findings of nutrition studies:

1. After certain minimum values of the nutrients are secured, additional quantities yield creasing (and, in some cases, eventually negative) returns to health.

2. The optimum quantity of any nutrient depends on the quantities of the other nutrients available.

Diminishing returns are illustrated by the fact that the amount of calcium in the body increases much more slowly than the input of calcium, and that increases of longevity are not proportional to calcium inputs. It appears possible in some cases to substitute one type of nutrition for another type. Stigler cites an example in which it was recommended that 30 micrograms of thiamine be substituted for 100 calories derived from sources other than fats. Another example cited is that a loss of riboflavin accompanies a deficiency of thiamine.

Stigler then turned to another question: How much of various nutrients are required? How do the requirements differ from one individual to another? In resolving this difficulty, he noted that the optimum quantity of calories is known only roughly or not at all. Many minima (to which 50 per cent is usually added as a safety factor) are found by determing the lowest level of input compatible with stable rate of loss of the nutrient through excreta. The interrelationships among various nutrients are even more obscure, and they are virtually ignored in dietary

recommendations. Even the statement of what substances are necessary for health is very complex. Thus, in addition to calcium, the body requires about 13 minerals (some in minute quantities), many kinds of vitamin, a dozen or so types of amino acids, and perhaps many more nutrients yet to be discovered.

The diets developed by Stigler were considerably lower in cost than those developed by others. One the reasons given is that the other diets included a greater variety of foods as a kind of “insurance” against omitting any of the unknown dietary elements. Another reason is that diet experts do give some weight to social and institutional pressures, particularly where they are not on firm grounds to support alternatives. On the other hand, Stigler justifies his diet in this regard by citing the National Research Council’s belief that there other minerals and vitamins are supplied in adequate amounts automatically when a certain group of common nutrients are obtained from natural foods. Based on considerations of this kind, the first step is setting up a mathematical model was to accept the Council’s statement of daily nutritional requirements.

Note that only nine of the more common nutrients were used, and he others were assumed to be automatically satisfied. It should also be noted that the requirements (discussed earlier) are rough, and, possibly with the exception of calories, almost any number in a very broad range probably could equally well be justified.

In considering the nutritive values of foods, again we see a similar situation, for the nutritive values of commons foods are known only approximately, and that is all that can be known about them. A large

margin of uncertainty arises on several scores. For example, the milligrams of ascorbic acid per 100 grams of apples vary with the type of apple:

Jonathan 4.4

McIntosh 2.0

Northern Spy 11.0

Ontario 20.8

Winesap 5.8

Winer Banana 6.6

The ascorbic acid in milk varies with season. Conditions of storage, such as temperature and length of time in storage, are important factors. The more corn matures, the greater is the amount of vitamin A, but the ascorbic acid content decreases. Long cooking decreases the nutritive value; well-done roasts of beef have roughly 70 percent of the thiamine, riboflavin, and miacin of raw cuts. Not only is there considerably variability in foods, which conceivably could be taken into account in programming, by introducing probabilistic considerations, but there is also the fact, according to Stigler, that the nutritive values that had been established in 1944 for many foods had been determined by obsolete and inaccurate techniques, or may be just plain wrong for other reasons.

Ignoring these difficulties, a model was set up in which some kind of average nutritive per unit quantity for each food as purchased was developed.

If x j units of the jth food were purchased and each food contained, per unit quantity, a i j units of the ith nutrient, then it was assumed that the individual would receive

∑j=1

n

aij x j

units of the ith nutrient, assuming there are no losses due to preparation of the foods. There is also a tacit assumption that there is no interaction between various foods; i.e., the total number of units of a nutrient available in a food is unaffected by the presence of some other foods in the diet.

Finally, a list of potential foods was selected for which retail prices were reported by the Bureau of Labor Statistics. The list was not complete since it excluded a lot all fresh fruits, many cheap vegetables rich in

nutrients, and fresh fish. Is these could have been included, it would seem that the minimum cost diet could be reduced by a substantial amount. However, other investigators have found that the optimal choice is quite insensitive to these particular prices due to the presence of certain staples in the optimum diet.

In table 27-1-II, the coefficients a ij per dollar expenditure are given for an abbreviated list of some 77 types of foods considered by Stigler. He recommended, however, where prices are subject to change because of local and seasonal conditions, and it is desired to compute not one but several such problems, that the units for measuring the quantity of foods be physical units such as weiht, or possibly volume in case of liquids. If this is done, the price data and the nutritive data can be developed independently.