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# chapter3 - 46 Chapter 3 Linear Programs A linear program...

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Chapter 3 Linear Programs A linear program involves optimization (i.e., maximization or minimization) of a linear function subject to linear constraints. A linear inequality con- straint on a vector x N takes the form a T x b or a 1 x 1 + a 2 x 2 + . . . + a N x N b for some a N and b . If we have a collection of constraints ( a 1 ) T x b 1 , ( a 2 ) T x b 2 , . . . , ( a M ) T x b M , we can group them together as a single vector inequality Ax b where A = ( a 1 ) T . . . ( a M ) T and b = b 1 . . . b M . When we write that the vector Ax is less than or equal to b , we mean that each component of Ax is less than or equal to the corresponding component of b . That is, for each i , ( Ax ) i b i . Sometimes, in a slight abuse of language, we refer to the i th row A i * of the matrix A as the i th constraint, and b i as the value of the constraint. In mathematical notation, a linear program can be expressed as follows: maximize c T x subject to Ax b. The maximization is over x N . Each component x j is referred to as a decision variable . The matrix A M × N and vector b M specify a set of M inequality constraints , one for each row of A . The i th constraint comes from the i th row and is ( A i * ) T x b i . The vector c N is a vector of values for each decision variable. Each c j represents the benefit of increasing x j by 1. The set of vectors x N that satisfy Ax b is called the feasible region. A linear program can also be defined to minimize the objective: minimize c T x subject to Ax b, 47
48 in which case c j represents the cost of increasing x j by 1. 3.1 Graphical Examples To generate some understanding of linear programs, we will consider two simple examples. These examples each involve two decision variables. In most interesting applications of linear programming there will be many more decision variables – perhaps hundreds, thousands, or even hundreds of thou- sands. However, we start with cases involving only two variables because it is easy to illustrate what happens in a two dimensional space. The situation is analogous with our study of linear algebra. In that context, it was easy to generate some intuition through two-dimensional illustrations, and much of this intuition generalized to spaces of higher dimension. 3.1.1 Producing Cars and Trucks Let us consider a simplified model of an automobile manufacturer that pro- duces cars and trucks. Manufacturing is organized into four departments: sheet metal stamping, engine assembly, automobile assembly, and truck as- sembly. The capacity of each department is limited. The following table provides the percentages of each department’s monthly capacity that would be consumed by constructing a thousand cars or a thousand trucks: Department Automobile Truck metal stamping 4% 2.86% engine assembly 3% 6% automobile assembly 4.44% 0% truck assembly 0% 6.67% The marketing department estimates a profit of \$3000 per car produced and \$2500 per truck produced. If the company decides only to produce cars, it could produce 22 ,

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chapter3 - 46 Chapter 3 Linear Programs A linear program...

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