Part 3 Linear Programming

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Part 3 Linear Programming

• 3.3 Theoretical Analysis

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Matrix Form of the Linear Programming Problem
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LP Solution in Matrix Form
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Tableau in Matrix Form
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Criteria for Determining A Minimum Feasible
Solution
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Theorem (Improvement of Basic Feasible Solution)

• Given a non-degenerate basic feasible solution
  with corresponding objective function f0, suppose
  for some j there holds cj-fjlt0. Then there is a
  feasible solution with objective value fltf0.
• If the column aj can be substituted for some
  vector in the original basis to yield a new basic
  feasible solution, this new solution will have
  fltf0.
• If aj cannot be substituted to yield a basic
  feasible solution, then the solution set K is
  unbounded and the objective function can be made
  arbitrarily small (negative) toward minus
  infinity.

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Optimality Condition

• If for some basic feasible solution cj-fj or rj
  is larger than or equal to zero for all j, then
  the solution is optimal.

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Symmetric Form of Duality (1)
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Symmetric Form of Duality (2)

1. MAX in primal MIN in dual.
2. lt in constraints of primal gt in constraints of
   dual.
3. Number of constraints in primal Number of
   variable in dual
4. Number of variables in primal Number of
   constraints in dual
5. Coefficients of x in objective function RHS of
   constraints in dual
6. RHS of the constraints in primal Coefficients
   of y in dual
7. f(xopt)g(yopt)

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Symmetric Form of Duality (3)
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Example
Batch Reactor B
Batch Reactor C
Batch Reactor A
Products P1, P2, P3, P4
Raw materials R1, R2, R3, R4
P1 P2 P3 P4 capacity time
A 1.5 1.0 2.4 1.0 2000
B 1.0 5.0 1.0 3.5 8000
C 1.5 3.0 3.5 1.0 5000
profit /batch 5.24 7.30 8.34 4.18
time/batch
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Example Primal Problem
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Example Dual Problem
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Property 1

• For any feasible solution to the primal problem
  and any feasible solution to the dual problem,
  the value of the primal objective function being
  maximized is always equal to or less than the
  value of the dual objective function being
  minimized.

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Proof
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Property 2
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Proof
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Duality Theorem

• If either the primal or dual problem has a finite
  optimal solution, so does the other, and the
  corresponding values of objective functions are
  equal. If either problem has an unbounded
  objective, the other problem has no feasible
  solution.

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Additional Insights
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Symmetric Form of Duality (3)
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LP Solution in Matrix Form
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Relations associated with the Optimal Feasible
Solution of the Primal problem
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Example
PRIMAL
DUAL
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Tableau in Matrix Form
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Example The Primal Diet Problem

• How can we determine the most economical diet
  that satisfies the basic minimum nutritional
  requirements for good health? We assume that
  there are available at the market n different
  foods that the ith food sells at a price ci per
  unit. In addition, there are m basic nutritional
  ingredients and, to achieve a balanced diet, each
  individual must receive at least bj unit of the
  jth nutrient per day. Finally, we assume that
  each unit of food i contains aji units of the jth
  nutrient.

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Primal Formulation
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The Dual Diet Problem

• Imagine a pharmaceutical company that produces in
  pill form each of the nutrients considered
  important by the dietician. The pharmaceutical
  company tries to convince the dietician to buy
  pills, and thereby supplies the nutrients
  directly rather than through purchase of various
  food. The problem faced by the drug company is
  that of determining positive unit prices y1, y2,
  , ym for the nutrients so as to maximize the
  revenue while at the same time being competitive
  with real food. To be competitive with the real
  food, the cost a unit of food made synthetically
  from pure nutrients bought from the druggist must
  be no greater than ci, the market price of the
  food, i.e. y1 a1i y2 a2i ym ami lt ci.

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Dual Formulation
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Shadow Prices

• How does the minimum cost change if we change the
  right hand side b?
• If the changes are small, then the corner which
  was optimal remains optimal. The choice of basic
  variables does not change. At the end of simplex
  method, the corresponding m columns of A make up
  the basis matrix B.

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