Chapter 8: Linear Programming Quantitative Decision Making 7th ed. By Lapin and Whisler

The story behind Linear Programming George B. Dantzig John von Neumann Leonid Kantorovich

Definition Linear programming is a mathematical method that is used to establish a plan that efficiently allocates limited resources to achievement of a desired objective. Linear programming is the process of maximizing or minimizing a linear function subject to a set of constraints.

Possible Applications for LP Developing a production schedule and inventory policy. Establishing a portfolio that maximizes return. Maximizing advertising effectiveness. Minimizing total transportation costs.

Main Characteristics of LP Problems Concerned with maximizing or minimizing some quantity. Restrictions or constraints that limit the degree to which the objective can be pursued. Only linear relationships are involved. http://en.wikipedia.org/wiki/Linear_programming

Three Components to LP Variables Constraints Objective Function Non-negativity conditions.

Example (p.263) The Redwood Furniture Company makes tables and chairs as part of its line of patio furniture. Resource Unit Requirements Amount Available Table Chair Wood(board ft) 30 20 300 Labor(hours) 5 10 110 Unit Profit $6 $8 How many tables and chairs should be made to maximize the total profit?

Redwood Furniture Problem Formulation Let XT and XC denote the number of tables and chairs to be made. (Define variables) Maximize P = 6XT + 8XC (Objective function) Subject to: (Constraints) 30XT + 20XC < 300 (wood) 5XT + 10XC < 110 (labor) where XT and XC > 0 (non-negativity conditions) Letting XT represent the horizontal axis and XC the vertical, the constraints and non-negativity conditions define the feasible solution region.

Feasible Solution Region for Redwood Furniture Problem

Graphing to Find Feasible Solution Region For an inequality constraint (with < or >), first plot as a line: 30XT + 20XC = 300. Get two points. Intercepts are easiest: Set XC = 0, solve for XT for horizontal intercept: 30XT + 20(0) = 300 => XT = 300/30 = 10 Set XT = 0, solve for XC for vertical intercept: 30(0) + 20XC = 300 => XC = 300/20 = 15 Above gets wood line. Do same for labor. Mark valid sides and shade feasible solution region. Any point there satisfies all constraints and non-negativity conditions.

Graphing to Find Feasible Solution Region To establish valid side, pick a test point (usually the origin). If that point satisfies the constraint, all points on same side are valid. Otherwise, all points on other side are instead valid. Equality constraints have no valid side. The solution must be on the line itself. Some constraint lines are horizontal or vertical. These involve only one variable and one intercept.

Finding Most Attractive Corner The optimal solution will always correspond to a corner point of the feasible solution region. Because there can be many corners, the most attractive corner is easiest to find visually. That is done by plotting two P lines for arbitrary profit levels. Since the P lines will be parallel, just hold your pencil at the same angle and role it in from the smaller P’s line toward the bigger one’s That is the direction of improvement. Continue rolling until only one point lies beneath the pencil. That is the most attractive corner. (Problems can have two most attractive corners.)

Most Attractive Corner for Redwood Furniture Problem

Finding the Optimal Solution The coordinates of the most attractive corner provide the optimal levels. Because reading from graph may be inaccurate, it is best to solve algebraically. Simultaneously solving the wood and labor equations, the optimal solution is: XT = 4 tables XC = 9 chairs P = 6(4) + 8(9) = 96 dollars Note: supply the computed level of the objective in reporting the optimal solution.

Finding most attractive corner algebraically Identify corner points: (0,0), (0,11), (10,0) (4,9) Substitute into objective function and compare values: P = 6XT + 8XC (XT=0, XC=0) P=6(0)+8(0)=0 (XT=0, XC=11) P=6(0)+8(11)=88 (XT=10, XC=0) P=6(10)+8(0)=60 (XT=4, XC=9) P=6(4)+8(9)=96

Feasible Solution Region for Redwood Furniture Problem