LINEAR PROGRAMMING
Classic Types of Constrained Decision
Problems
1. Product Mix
2. Ingredient Mix
3. Transportation
4. Assignment
5. Time Period
RECOGNITION
CHARACTERISTICS OF LP PROBLEMS
1. Single, well-defined objective
2. Alternative courses of action
3. Achievement of the objective must be
constrained by scarce resources or other
restraints
4. The objective and each of the constraints
must be expressible as linear, mathematical
functions
FORMULATION
Max Z= objective fn. eq.
(Min)
s.t.
constraint eq. 1
constraint eq. 2
etc.
nonnegativity condition
FORMULATION
EXAMPLE
Max Z= 5x +2y
s.t.
10x +4y < 445
3x + 2y > 230
5x - 4y = 125
x,y > 0
LP TERMINOLOGY
• Objective fn.--the mathematical function
representing our objective which is being
optimized.
• Constraints--restrictions on our solution
• Decision variables--variables of the obj. fn.
and constraint eqs.
• Feasible solution space--solutions for which
all the constraints are satisfied
LP TERMINOLOGY
• Optimal Solution--a feasible solution that
has the most favorable value of the
objective function.
GRAPHICAL LP SOLUTION
OUTLINE
•
•
•
•
Formulate objective fn. and constraint eqs.
Plot constraints--treat as equalities initially
Determine feasible solution space
Determine optimal solution
--Enumeration Method
--Objective Fn. Family of Parallel Lines
ENUMERATION METHOD
• Relies on the fact that feasible solution
space is a CONVEX SET
• Identify corner (kink) points on the outer
boundary of the feasible solution space
• Plug coordinates of corner points into
objective function
• Select corner point with the best obj. fn.
value (z) as the optimal solution point
OBJECTIVE FUNCTION
FAMILY OF PARALLEL LINES
• Assume a value for Z and plot that member
of the family
• Assume another value for Z and plot it
• Note the lines are parallel to each other
• If maximizing, find the highest member of
this family of lines just tangent to a final pt.
in the feasible space. It is the optimum.
• If minimizing, find the smallest member just
tangent , it is the optimum.