LrChVIISIILinearProgramming052 - Qualitative Choice...

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Qualitative Choice Analysis Workshop 1 Econometrics Laboratory
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7.2 Linear Programming Dr. Raja Mohammad Latif Objective: To state the nature of a linear programming problem, to introduce terminology associated with it, and to solve it geometrically.
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Dr. Raja Latif, Math 131-052 (Feb 12-10June, 2006) 3 1 Chapter 7: Linear Programming. 7.2: LINEAR PROGRAMMING Abstarct: We will learn to state the nature of a linear pro- gramming problem along with the introduction of terminology associated with it, and then developing a method for its solution geometrically. Many business and economic problems are concerned with optimizing (maximizing or minimizing) a function subject to a system of equalities or inequalities. The function to be optimized is called the objective function cost functions are examples of objective functions. The system of equalities and inequalities to which the objective function is subjected re±ects the constraints (for example, limitations on resources such as materials and labor) imposed on the solution(s) to the problem. Problems of this nature are called mathematical programming problems . In particular, problems in which both the objective function and the constraints are expressed as linear equations or inequalities are called linear programming problems. A linear programming problem consists of a linear objective function to maximized or minimized subject to certain Department of Mathematics, KFUPM
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Dr. Raja Latif, Math 131-052 (Feb 12-10June, 2006) 4 constraints in the form of linear equalities or inequalities. Existence of a Solution. Consider a linear programming problem with the set R of feasible points and objective function z = Ax + BY: 1 : If R is bounded, then z has a maximum and a minimum value on R: 2 : If R is unbounded and A & 0 ;B & 0 ; and the constraints include x & 0 and y & 0 ; then z has a minimum value on R but not a maximum. 3 . If R is the empty set, then the linear programming problem has no solution and z has neither a maximum nor a minimum value. Fundamental Theorem of Linear Programming If a linear programming problem has a solution, it is located at a corner point of the set of feasible points. If a linear programming problem has multiple solutions, at least one of them is located at a corner point of the set of feasible points. In either case the corresponding value of the objective function is unique. Steps for Solving a Linear Programming Problem
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LrChVIISIILinearProgramming052 - Qualitative Choice...

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