By Quirino Paris
This textual content covers the elemental conception and computation for mathematical modeling in linear programming. It offers a powerful history on tips on how to arrange mathematical proofs and high-level computation equipment, and contains gigantic heritage fabric and path. Paris offers an intuitive and novel dialogue of what it capacity to resolve a approach of equations that may be a an important stepping stone for fixing any linear application. The dialogue of the simplex technique for fixing linear courses provides an financial interpretation to each step of the simplex set of rules. The textual content combines in a special and novel manner the microeconomics of construction with the constitution of linear programming to provide scholars and students of economics a transparent suggestion of what it capability, formulating a version of financial equilibrium and the computation of chance price within the presence of many outputs and inputs.
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Additional info for An Economic Interpretation of Linear Programming
Total input use is deﬁned as the input requirement per unit of output multiplied by the output level and summed over all products. The constraints of problem (P) represent the “production function” of economic theory in an unfamiliar speciﬁcation. In order to make the necessary connection, the production function is usually stated in terms of outputs as a function of inputs, that is, outputs = f (inputs). In linear programming, the “production function” is called production technology and is stated in a reciprocal form as h(outputs) = inputs.
Before deriving a dual problem, therefore, it is essential to state the primal problem in the form of a Max and ≤ constraints (or, alternatively, in the form of a Min and ≥ constraints). By strictly following this rule, the dual problem will be correctly set up and will be easily interpreted. In this vein of thought, it is also convenient to associate an objective function to be maximized with less-than-or-equal (≤) constraints and a minimization with greater-thanor-equal (≥) constraints. One further look at the speciﬁcation of the primal and dual problems (P ) and (D ): the transpositions indicated above, the structure of the constraints, 28 Chapter 1 and the independent appearance of the variables constitute a symmetry that plays a fundamental role in linear programming.
The land coeﬃcients are usually chosen to be of unit value, implying that all the other coeﬃcients for the same activity, say wheat, are measured with reference to a unit of land, say an acre. In this ﬁrst example, we deliberately choose a non-unit coeﬃcient for land to alert the reader to the important problem of measurement units, which will be further discussed in chapter 3. 45. Similarly, the coeﬃcient 4 in the constraints of problem (P ) indicates that for producing one unit of wheat it is necessary to use 4 units of labor.