COB 291 An Introduction to Management Science Dr. Mike Busing College of Business, James Madison University

Agenda Syllabus Review Algebra Review Quiz Introduction to Models/Modeling Introduction to Linear Programming (LP) Graphical Solution to LP

Announcements Please purchase and bring to next and subsequent classes: –Pack of graph paper –Pack of colored pencils –Straight edge or ruler

Linear Programming Deterministic Modeling LP problems contain uncontrollable variables The uncontrollable variable quantities are known The uncontrollable variable quantities are fixed or constant in short-run Used to solve problems where there is no uncertainty about the future.

Linear Programming Deterministic Modeling (cont’d) The GOAL of LP is to identify the decision that gives me the best outcome. However, there are millions of decision possibilities LP is a searching mechanism that sifts through all the possible (feasible) solutions to find the “best” solution. LP is a very efficient search technique

LP Example BMW must produce cars such that they satisfy the constraints of the production plan, the marketing plan, the finance plan, etc. In addition, this plan should generate the most profit, given the constraints of the various functional business areas. “x” million 3 series “x” million 5 series “x” million 7 series “x” million 8 series “x” million Z series “x” million SUV

LP Example Advertising: Limited amount of money to invest in different advertising mediums “x” million dollars in radio spots “x” million dollars in television spots “x” million dollars in newspaper ads “x” million dollars in magazine ads “x” million dollars in billboards

LP Example Finance: As a mutual fund manager, you must take all the money and invest it in various instruments “x” million dollars in stocks “x” million dollars in fixed income bonds “x” million dollars in money funds “x” million dollars in annuities “x” million dollars in cash

LP Class Example - Seuss’s Sandwich Shop Seuss’s Sandwich Shop sells two types of sandwiches: green eggs and ham (GEH) and ham and cheese (HC). A green eggs and ham sandwich consists of 2 slices bread, 1 green egg, and 1 slice ham. It takes an employee 3 minutes to make one of these sandwiches. A ham and cheese sandwich consists of 2 slices bread, 2 slices ham, and 1 slice cheese. It takes 2 minutes to make a ham and cheese sandwich. The Sandwich Shop presently has available 400 slices of bread, 80 slices of cheese, 120 green eggs and 200 slices of ham. The shop also has one employee scheduled for 7 hours to make all of the sandwiches. If a green egg and ham sandwich sells for $5 and a ham and cheese sandwich sells for $4, how many of each type should be prepared to maximize sales revenue? (Assume that demand is great enough to ensure that all sandwiches made will be sold.)

LP Class Example - Seuss’s Sandwich Shop We will figure out how to represent the English sentences in the problem via convenient mathematical equations. This is the most important part of linear programming.

LP Class Example - Seuss’s Sandwich Shop 3 steps in all LP formulations: 1st: Decision Variables - why am I solving this problem? - what are the unknown quantities in the problem? - the minimum number of decision variables is 2 and the maximum is several hundred thousand in practical situations.

LP Class Example - Seuss’s Sandwich Shop 3 steps in all LP formulations: 2nd: Objective Function - what is the motivation behind this problem? (e.g., maximize profit or minimize cost?) 3rd: Constraints - issues that force decision variables from taking on -  to +  values. - no restriction on number of constraints. - non-negativity is extremely important.

LP (cont’d) So why do we call it “Linear” Programming?

LP Class Example - Seuss’s Sandwich Shop Steps: 1. Identify the objective function, z. 2. Identify the constraints. 3. Identify what we mean by “feasible solution” 4. Identify the “optimal solution” 5. Identify the “binding constraints”

LP Class Example - Seuss’s Sandwich Shop

GRAPHICAL SOLUTION TO PROBLEM