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# TrChVIISII - 1 Chapter 7 Linear Programming 7.2 LINEAR...

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1 Chapter 7: Linear Programming. 7.2: LINEAR PROGRAMMING Abstarct : We will learn to state the nature of a linear programming problem along with the intro- duction 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 func- tion subject to a system of equalities or inequalities. The function to be optimized is called the objective function . Pro°t functions and cost functions are examples of objective functions. The system of equalities and inequalities to which the objective function is subjected re±ects the con- straints (for example, limitations on resources such

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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 sub- ject to certain constraints in the form of linear equal- ities or inequalities. Existence of a Solution. Consider a linear program- ming 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 program- ming problem has no solution and z has neither a maximum nor a minimum value. Fundamental Theorem of Linear Programming

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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 (SM175) If a linear programming problem has a solution, fol- low these steps to °nd it: STEP 1 Write an expression for the quantity that is

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TrChVIISII - 1 Chapter 7 Linear Programming 7.2 LINEAR...

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