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### 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 ProgrammingDeterministic Modeling

Used to solve problems where there is

no uncertainty about the future.

- LP problems contain uncontrollable variables

- The uncontrollable variable quantities are known

- The uncontrollable variable quantities are fixed or
- constant in short-run

Linear ProgrammingDeterministic 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

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