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

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

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

Deterministic Modeling
• LP problems contain uncontrollable variables
• The uncontrollable variable quantities are known
• The uncontrollable variable quantities are fixed or constant in short-run

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

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

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

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.

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
# of Green Eggs and Ham Sandwiches
# of Ham and Cheese Sandwiches
Sales Revenue $ -
ITEM
Bread
Cheese
0
0
Green Eggs and Ham Ham and Cheese
Green Eggs
Ham
Available
400
80
120
200
Used
0
0
0
0

LP Class Example - Seuss’s Sandwich Shop
GRAPHICAL SOLUTION TO PROBLEM