L04f05

L04f05 - Chapter 3: Standard Forms 1 Chek Beng Chua...

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Unformatted text preview: Chapter 3: Standard Forms 1 Chek Beng Chua Introduction LP problems takes many form three types of constraints and two types of objectives. Instead of developing theories and methods for all kinds of LP, we focus on TWO simple forms of LP problems Standard Inequality Form (SIF) and Standard Equality Form (SEF). We shall show that all LP problems is equivalent to one in SIF and one in SEF. Chapter 3: Standard Forms 2 Chek Beng Chua Definitions Standard Inequality Form (SIF) maximize c T x subject to Ax b x = non-negativity constraints 1. It is a maximization problem. 2. All variables are required to be non-negative . These constraints are called non-negativity constraints . 3. All other constraints are linear inequalities of the type . Example The factory production problem is in SIF. maximize n j =1 c j x j subject to n j =1 a ij x j b i ( i = 1 , 2 , . . . , m ) x j ( j = 1 , 2 , . . . , n ) Chapter 3: Standard Forms 3 Chek Beng Chua Standard Equality Form (SEF) maximize c T x subject to Ax = b x 1. It is a maximization problem. 2. All variables are required to be non-negative . 3. All other constraints are linear equations . Example A reformulation of the transportation problem is in SEF. (Compare with LP on page 12.) maximize- p i =1 q j =1 c ij x ij subject to q j =1 x ij = s i ( i = 1 , 2 , . . . , p ) p i =1 x ij = t i ( j = 1 , 2 , . . . , q ) x ij i = 1 , 2 , . . . , p, j = 1 , 2 , . . . , q Chapter 3: Standard Forms 4 Chek Beng Chua Example The problem of an overdetermined system of linear equa- tions is not in SIF nor is it in SEF. minimize m i =1 y i subject to n j =1 a ij x j- b i- y i ( i = 1 , 2 , . . . , m ) n j =1 a ij x j- b i + y i ( i = 1 , 2 , . . . , m ) Fortunately, this LP problem and all LP problems for that matter can be transformed into an equivalent LP in standard form (i.e., SEF or SIF). Chapter 3: Standard Forms 5 Chek Beng Chua Transformations We shall show that Every LP problem is equivalent to some LP problem in SIF (or SEF).problem in SIF (or SEF)....
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L04f05 - Chapter 3: Standard Forms 1 Chek Beng Chua...

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