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## Duality in Linear Programming

### Linear Programming in Industry (1960-01-01): 90-95 , January 01, 1960

Linear programming models possess the interesting property of forming pairs of symmetrical problems. To any maximization problem corresponds a minimization problem involving the same data, and there is a close correspondence between their optimal solutions. The two problems are said to be “*duals*”of each other.

## Front Matter - Linear Programming in Industry

### Linear Programming in Industry (1960-01-01) , January 01, 1960

## A Practical Example

### Linear Programming in Industry (1960-01-01): 15-25 , January 01, 1960

Linear programming methods have been successfully used for solving many industrial *blending problems.* The simple example in Ch. I belongs to this class of problems and so does the following practical example from the ice cream industry^{1}.

## Computational Procedures for Solving Linear Programming Problems

### Linear Programming in Industry (1960-01-01): 65-89 , January 01, 1960

As we have seen in Ch. II^{2}, the simplex procedure can be described as a systematic method of examining the set of basic feasible solutions, starting in an arbitrary initial basis of *m* variables (activities) where m is the number of linear restrictions. If the initial basic solution does not satisfy the simplex criterion, we move to a neighboring basis by replacing one of the basic variables, and so forth, until a basic feasible solution is attained in which all of the simplex coefficients are non-positive (in a minimization problem, non-negative). By the Fundamental Theorem, such a solution will be an optimal solution.

## The Applicability of Linear Programming in Industry

### Linear Programming in Industry (1960-01-01): 100-103 , January 01, 1960

The linear programming model is particularly suited for solving *short-run* problems, i.e., problems of operations planning which do not involve investment decisions. A typical example is the planning of production under given capacity limitations. The period for which operations are to be planned is comparatively short so that the fixed equipment does not undergo any changes; the technological and economic restrictions on the company’s freedom of action remain the same during the period, and the production program will have to respect these limitations.

## Elements of the Mathematical Theory of Linear Programming

### Linear Programming in Industry (1960-01-01): 5-14 , January 01, 1960

The general problem of linear programming can be formulated as follows: Find a set of numbers *x*_{1}, *x*_{2},.., *x*_{n} which satisfy a system of linear equations (side conditions)
(1a)
$$\begin{array}{*{20}{c}}
{{a_{11}}{x_1} + {a_{12}}{x_2} + .... + {a_{1n}}{x_n} = {b_1}} \\
{{a_{21}}{x_1} + {a_{22}}{x_2} + .... + {a_{2n}}{x_n} = {b_2}} \\
{....} \\
{{a_{m1}}{x_1} + {a_{m2}}{x_2} + .... + {a_{mn}}{x_n} = {b_m}}
\end{array}$$
and a set of sign restrictions (non-negativity requirements)
(1b)
$${x_1} \geqslant 0,{\kern 1pt} {\kern 1pt} {x_2} \geqslant 0,{\kern 1pt} ....,{\kern 1pt} {x_n} \geqslant 0$$
and for which the linear function
(1c)
$$f = {c_1}{x_1} + {c_2}{x_2} + .... + {c_n}{x_n}$$
has a maximum.

## Back Matter - Linear Programming in Industry

### Linear Programming in Industry (1960-01-01) , January 01, 1960

## Industrial Applications

### Linear Programming in Industry (1960-01-01): 33-64 , January 01, 1960

One of the first practical problems to be formulated and solved by linear programming methods was the so-called *diet problem*, which is concerned with planning a diet from a given set of foods which will satisfy certain nutritive requirements while keeping the cost at a minimum. For each food the nutritional values in terms of vitamins, calories, etc. per unit of food are known constants and these are the a’s of the problem, aii being the amount of the *i*th nutritional factor contained in a unit of the *j*th food. If it is required that there shall be at least b_{i} units of the *i*th nutrient in the diet the nutritional requirements will take the form of a set of linear inequalities’ in the variables *x1*, which represent the amounts of the respective foods which shall be present in the diet. These restrictions will in general be satisfied by a large number of combinations of ingredients (foods) and we want to select a combination which minimizes the total cost of ingredients, i.e., a linear function in the *x*_{j} where the coefficients c_{j} are the prices per unit of the respective foods.

## The Effects of Coefficient Variations on the Solution

### Linear Programming in Industry (1960-01-01): 96-99 , January 01, 1960

There are three groups of parameters in a linear programming problem: the “technological” coefficients, *a*_{ij} (representing, for example, machine time per unit of product); the constant terms on the right-hand sides of the restrictions, *b*_{i} (e.g., capacity limits); and the coefficients in the linear preference function, *c*_{i} (for example, unit profits). In practical applications of linear programming it is important to explore the *sensitivity* of the numerical solution with respect to changes in these parameters^{1,2}. Some of them may be subject to known variations in time—prices or cost elements change, machine times are reduced and capacities increased because of rationalization or technological change, output stipulations vary from period to period, etc.—or it may not be possible to determine them exactly but only within certain intervals. (When these variations are of a random nature, the coefficients should be thought of as probability distributions rather than numbers and the problem becomes a *stochasticprogramming problem.*)