Chapter 1 Linear programming problems 1.1 Linear programming problems Example 1.1. A feed supplier wishes to formulate a feed for chickens. Each chicken requires at least 250, 150, and 400 units of certain nutritive elements, which we denote by A,B, and C. There are four possible materials 1, 2, 3, and 4, to blend the feed from. Each material has a different cost and different nutritional values, as given in the table below. Nutritive Elements Feed material 1 2 3 Min. Daily requirement 4 A B C unit cost 3 1 2 20 15 25 15 25 20 10 30 250 20 25 20 30 4 150 400 We want to determine the cheapest feed that can be made that satisfies the minimal nutritional daily requirements. Find a mathematical formulation of this problem. 1
Definition 1.2. If C1, C2, ... Cn E R, then a function f(x1, x2, ... , In) in variables x1, x2, ... , In is a linear function if it is of the form n f(x1, x2, . . . , In ) = C101 + C202 + ... + CnIn = ? Cjaj. j=1 Example 1.3. The following are linear functions • . 5x1 + 6x4 - Tx2. The following are not linear functions · x2 . x1 + 6x4 - x2 + x 3 . x1x2. Definition 1.4. Let f(x1, x2, ... , In) be a linear function and b be a real number. Then f(x1,@2, ... , In) = b is a linear equality; and are linear inequalities. f(x1, x2, ... , In) ?b and f(x1, x2, . .. , In ) ? b If an equation/inequality can be rewritten as a linear equation/inequality then we will also call it a linear equation/inequality. Example 1.5. · x3 ? 0 is a linear inequality. . x1/ 2 ? 4 is a linear inequality as it can be written as x1- 4x2 ? 0. . 1/ 2 + x3 ? 4 is not a linear inequality. 2
1.1. LINEAR PROGRAMMING PROBLEMS Definition 1.6. A Linear programming problem (or LPP) is an optimisation problem for which: 1. We attempt to maximise or minimise a linear function z(x1, x2, ... , In), which is called the objective function. 2. The variables x1, x2, ... , In satisfy a set of constraints, each of which is a linear inequality or a linear equality. The variables x1, ... , Im are called the decision variables. The constraints usually include sign restrictions on the variables: Example 1.7. xi >0, or xi ?0, or xi is unrestricted. Minimize subject to 3x1-22 objective function ?-3 <- lin. ineq. -x1+6x2-3 + 14 7x2 + 2x4 = 5 « lin. eq. x3 ? 2 «- lin. ineq. x2, 13 ?0 sign restrictions The decision variables here are x1, x2, 13. Example 1.8. Maximize subject to 5x1+ 4x2 + 3x3 2x1+ 3x2 + X3 ? 250 4x1+x2 + 2x3 ? 11 3x1 + 4x2 + 2x3 x1, x2, 13 ? 8 ?0. Definition 1.9. A