Linear Programming

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Linear Programming
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Formulating the Linear Programming Problem

• Linear optimization (more widely known as linear
  programming) involves linear functions only.
• It can be analyzed using only the methods of
  linear algebra.
• Classic problem a minimization problem and a
  maximization problem.

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Example. A Nutrition Aid Problem

• A Nutrition Aid Program is considering two food
  supplements S1 and S2 for distribution among a
  population deficient in nutrients N1, N2, and N3.
  The nutritional content and prices of the food
  supplements and the minimum daily nutritional
  requirements per person are given in Table 3.1
• For example, each unit of S1 provides 4 units of
  N1, 8 units of N2, 5 units of N3, and costs 6
  pesos.
• The problem is to determine the daily quantities
  of the food supplements for each person that meet
  the minimum daily requirements and entail the
  least cost.

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Table 3.1. Dietary and Price Data
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Example. A Product Mix Problem

• A firm produces two different products P1 and P2.
  Every unit of P1 uses 3 units of resource 1 (R1)
  and 1 unit of resource 2 (R2) and earns a profit
  of 5 pesos. The firm wants to determine how many
  of the two products must be manufactured each
  week in order to maximize profit subject only to
  resource availability.

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Forms of LP problems

• Each model has a linear function to be optimized
  (maximized or minimized) this function is called
  the objective function.
• The optimization of the objective function is
  subject to linear inequality constraints either
  of the type "" or of the type .
• The constraints also include nonnegativity
  restrictions on the variables.
• In its most general form, a linear programming
  problem can contain both types of inequality
  constraints as well as equality constraints.
  Moreover, the variables need not be constrained
  to be nonnegative

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General Form of the LP
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Notes

• In solving an LP, the constraints (except the
  nonnegativity of the variables), are transformed
  into equalities.

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Note

• Since max f(x) -min (-f(x)), then any
  maximization problem may be replaced by a
  minimization problem by changing "Max f(x)"to
  "Min f(x)". Similarly, a minimization problem
  may be replaced by a maximization problem by
  changing "Min f(x)" to "Max f(x)".

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Standard Form of the LP
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Definitions

• Feasible solution - a vector x which satisfies
  the constraints of an LP. The set of feasible
  solutions is called the feasible set or feasible
  region. The value of the objective function at a
  feasible solution x is denoted by VOF(x).
• Optimal Solution. A feasible solution that makes
  the value of the objective function an optimum
  (maximum or minimum) is called an optimal
  solution (maximizer, minimizer).

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No feasible solution
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No optimal solution
Feasible region is unbounded
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Assumptions of the LP

• The system Ax b has a solution. (The case where
  there is no solution is of no interest).
• The system Ax b is non-redundant, i.e., no row
  of Ab is a linear combination of the others
  hence, the row vectors of Ab are linearly
  independent. This is equivalent to the condition
  that rank(Ab) m. Hence, rank(A) m. This
  implies that m lt n since the rank of a matrix
  cannot exceed the number of columns.
• The system Ax b has more than one solution.
  This implies
• m lt n. For, if m n, then A is an n x n matrix
  and, by assumption rank(A) n, i.e, A is
  nonsingular which would imply that the equation
  Ax b has a unique solution.
• The system Ax b has an infinite number of
  solutions. For, if x and y are two distinct
  solutions, then ax (1 - a)y is also a solution
  for all a such that 0 lt a lt 1.

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Graphical Solution
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Optimal Allocation of Resources
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Simplex method overview

• An optimal solution may occur only on the
  boundary of the feasible region. For if x1 is an
  interior point of the feasible region, then the
  profit line through x1 can be moved to another
  feasible point x2 with a higher profit, showing
  that x1 cannot be optimal.

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Simplex method overview

• An optimal solution can be a corner of the
  feasible set, or a noncorner point of the
  boundary
• Theorem. If an LP has an optimal solution, then
  it has an optimal solution that is a corner
  (extreme point, vertex) of the feasible set.

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• Profit lines are parallel to the boundary AB.
  Optimal solutions coincide with AB
• All points are optimal, including endpoints which
  are corners of a feasible set.
• Important whenever an optimal solution exists,
  there is always one that is a corner of a
  feasible set.

Theorem. If an LP has an optimal solution, then
it has an optimal solution that is a corner
(extreme point, vertex) of the feasible set.
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Simplex method overview

• The Theorem suggests that searching for an
  optimal solution need not include all feasible
  solutions. The search can be confined only to the
  corners of the feasible set.

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Simplex method overview

• The Theorem suggests that searching for an
  optimal solution need not include all feasible
  solutions. The search can be confined only to the
  corners of the feasible set.
• Enumerating all the corners of the feasible set
  to obtain the optimal solution is not a very
  practical method since many linear programming
  problems could have millions of corners.
• The Simplex Algorithm is a systematic procedure
  to find the optimal solution.