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.