UNIT 3 An introduction to Linear Programming An LP problem An - - PowerPoint PPT Presentation

UNIT 3

An introduction to Linear Programming

An LP problem

An engineering factory makes 4 products (PROD1 to PROD4) on the following machines: 4 grinders, 2 drills and 1 planer. Each product yields a certain contribution to profit (defined as selling price in €/unit minus cost of raw materials). These quantities together with the unit production times (hours) required on each process are given in the table below. PROD 1 PROD 2 PROD 3 PROD 4 Contribution to profit 10 6 8 4 Grinding 0.5 0.7

• Drilling

0.1 0.2

• 0.3

Planing

• 0.01
• If there are 8 working hours in a day, what should the factory

produce (daily) in order to maximize the total profit?

An LP model

Canonical form of a Linear Programming (LP) problem

,..., , ... ... ... ... . . ...

2 1 1 1 2 2 1 21 1 1 1 11 1 1

≥ ≤ + + ≤ + + ≤ + + + +

n m n mn m n n n n n n

x x x b x a x a b x a x a b x a x a t s x c x c Max

If it is an LP maximization problem, all the constraints must be ≤ All the variables must be non-negative If it is a minimization problem, the constraints must be ≥

Canonical form of a Linear Programming (LP) problem

. . ≥ ≤ x b x A t s x c Max

t

Technical matrix Coefficients vector RHS vector

: : : b c A

Standard form of an LP problem

. . ≥ = x b x A t s x c Max

t

All the constraints must be =

An LP problem can be transformed into its standard form adding slack variables.

Characteristics of LP problems



S is always a convex set (but it may be bounded or unbounded).



An LP problem can be feasible and not have an optimal solution (it can be unbounded) or be bounded and not have an optimal solution (infeasible).



All the optima (if any) are global optima (because the Local-Global theorem can always be applied).



A solution is optimal if and only if it is a K-T point.



Optimal solutions are always boundary (border) points (never interior points), and at least one of them will be a vertex (corner point) of S.



If two solutions are optimal, all the solutions in the segment that joins them are

• ptimal too.

Basic feasible solutions (BFS)

Definition: A solution is basic if:

• .
• n-m of its elements are zero.
• The submatrix of A associated with basic variables (basic matrix) is

regular (its determinant is not zero).

• All the variables satisfy the non-negativity constraints

x

b x A =

Non-basic variables The remaining variables, which can take non-zero values A basic feasible solution is called degenerate if any of its basic variables takes value 0.

Example

, 5 8 2 . . 5 4 ≥ ≤ ≤ + + y x y y x t s y x Max

, , , 5 8 2 . . 5 4 ≥ = + = + + + t s y x t y s y x t s y x Max

        = 1 1 1 1 2 A t s y x

m=2 equations n=4 variables Technical matrix

Example

 Is a BFS?

) 5 , 8 , , ( = x

• 2 basic variables: s,t (2 non-basic variables: x,y)
• Basic matrix
• 

       =                       = 5 8 5 8 1 1 1 1 2 x A

1 1 1 ≠ =         = B B

, , , ≥ t s y x

Basic feasible solution

Obtaining BFSs

 Given a set of basic variables, if |B| is not 0, the value of

these basic variables can be obtained (if |B|=0, there is no basic feasible solution associated with this set of basic variables).

Obtaining BFSs: example

, , , 5 8 2 . . 5 4 ≥ = + = + + + t s y x t y s y x t s y x Max

1 1 1 1 ≠ =         = B B

        =                 5 8 1 1 1 t y

3 , 8 5 8 − = =    = + = t y t y y

Basic variables: y,t (0,8,0,-3) is not a basic feasible solution, since t<0.

Obtaining BFSs: example

, , , 5 8 2 . . 5 4 ≥ = + = + + + t s y x t y s y x t s y x Max

1 2 =         = B B

Basic variables: x,s There is no basic feasible solution with the basic variables x,s

Obtaining BFSs: example

, , , 5 8 2 . . 5 4 ≥ = + = + + + t s y x t y s y x t s y x Max

2 1 1 2 =       = B B

Basic variables: x,y (5,1.5,0,0) is a BFS

      =             5 8 1 1 2 y x 2 3 , 5 5 8 2 = =    = = + x y y y x

Fundamental theorem of Linear Programming

 If an LP problem has at least one feasible solution,

then it will also have at least one BFS.

 If an LP problem has optimal solution(s), then at

least one optimal solution will be a BFS.