Linear Programming

1. A Conceptual introduction Linear Programming

2. Mathematical Programming Some business decisions (or components) can be framed as an �optimization problem�, i.e., There are quantitative decision variables that represent our decision (e.g., what price p � in dollars � should I ask per dozen donuts in a fund raiser?) There is a quantitative objective function based on the decision variables which calculates �how good� a decision is (e.g., what profit can our fund raiser expect if the per dozen donut price is p?) There may be constraints which limit the possible values of decision variables (e.g., above $10 per dozen no one will purchase donuts even for a good cause so p=10 is required.) There are analytic tools (maybe pencil & paper but more likely computer software) which can find values of the decision variables which maximize (or minimize) the objective.

3. A Classic Non-Linear Program �

4. Key: A Business Decision is Modeled Programmatically

5. Linear Programming Problems where the objectives and constraints are linear turn out to be relatively easy to solve If x1, x2, �, xn are the (decision) variables then linear functions are the form of a1�x1 + a2�x2 + � + an�xn Linearity is equivalent to proportionality and additivity: F(2�x) = 2�F(s) (e.g., twice as many cost twice as much) F(x+y) = F(x) + F(y) (e.g., the cost of 3 burgers and two orders of fries together is the same as the cost of 3 burgers plus the cost of two orders of fries Widely applied in industry; Problems with thousands to millions of variables Often LP is embedded in software (e.g., scheduling, ERP)

6. LP Formulation

7. White Company (From the Book)

8. LP solution Graphical approach for trivial, two variable problems Shows issues Not a general purpose method at all Excel Solver for class oriented problems Special purpose or embedded solvers for industrial strength problems

9. White Company Graphical Solution

10. Case Study: Kendall Corporation Augusta plant manufactures variety of gauze bandages & sponges Processes: Greige rolls received as raw materials � about 54� wide & 25,000 yds long (if unrolled) Rolls are re-rolled to even up tension Rolls are boiled in bleach solution in kiers Rolls are spun to begin drying & distribute water Rolls are dried in tenters As cloth exits the tenter, rolls are cut into usable widths Slit rolls are work-in-process inventory used in making finished bandages & sponges

11. Kendall Distribution

12. A Kendall Toy Problem Tentered gauze in a style to be scheduled can be stretched to between 15� and 18� Slit rolls needed today are 28 4� rolls, 34 7� rolls, and 22 10� rolls From these inputs a software program Finds all �legal� slit roll combinations Formulates a LP Decision variables - # rolls to slit at each combination Objective � Minimize total waste or total product consumed or total inventory produced in slitting rolls Constraints � Ensure that demand for each roll width is met Our �toy� problem has 5 variables & three constraints; typical Kendall problem is about 1500 variables and 20 constraints

13. A Conceptual introduction Linear Programming