1. Introduction

1
1. Introduction

• Steps in Problem Solving and Decision Making
• Identify the Problem
• Determine Alternatives
• Determine the Criteria to evaluate alternatives
• Evaluate Alternatives
• Choose an alternative
• Implement the chosen alternative
• Evaluate the Results

2
Models

• Models are representations of reality.
• We interested in mathematical models.
• Why mathematical modeling?
• Can understand a real world phenomenon
• better, quicker.

3
Well Known Models

• Break-even Analysis
• Annual Fixed Costs 100,000
• Price /Unit 50
• Variable Costs/Unit 30
• Break-even Point in Units 100,000 / (50-30)
• 5,000 units
• BEP in 5,000 50
• 250,000

4
Well Known Models

• Capital Budgeting
• Present Value of outflows 545,000
• Present Value of Inflows 700,000
• Net Present Value 155,000

5
Do Models Have to be Realistic?

• BEP
• Are revenues linear in of units?
• Quantity discounts
• Are all costs fixed and variable?
• Step costs
• What happens to the inventories?
• Realism of models is desirable, but not
  necessary. Criteria is does the model help us
  understand, predict, control the real world
  situations better

6
Constants, Variables, Linearity

• A variable can take different values.
• P price per unit
• V Variable cost per unit
• F Fixed cost per year
• BEP F / (P-V)
• F, P, V are variables. We can assign different
  values
• F 9 / 5 C 32
• F Fahrenheit, C centigrade Variables
• 9, 5, 32 are constants.

7
Constants, Variables, Linearity

• A function is linear in a variable if the
  variable is multiplied (or divided) by a constant
  and a constant is added.
• y a b x
• a, b are constants. Y is linear in x since it is
  multiplied by a constant and a constant is added.
• y bx and Y a x are linear
  functions
• Y x2, y log (x), y wx where w is another
  variable are not linear functions.

8
Constants, Variables, Linearity

9
Linear Functions

• b
  (slope)
• a (Intercept)
• Equation of the line is y a bx

10
Slope

• Increase in y for unit increase in x.
• If (2,12) and (5,24) are two points on a line,
  then for 3 (5-2) units of increase in x, y has
  increased by 12 (24-12)
• Slope 12 /3 4.
• Plot y 2x, y -3x, y -5 3x
• Y -12 -4x

11
Linear Functions

• Determine intercept (a), Slope (b) and the
  equation of the line
• 5
• 
  
• 
  15

12
Linear Functions Determine intercept (a), Slope
(b) and the equation of the line

6
-15
13
Linear Functions Determine intercept (a), Slope
(b) and the equation of the line

10
-4
-10
14
Decision Variables, Parameters
Uncontrollable Inputs Parameters
Mathematical Model
Solution
Controllable Variables Decision Variables
15
Alternatives and Criterion in Decision Making

• Criterion is the objective of decision making.
• A firm may be interested in picking a price to
  maximize revenues, profits, customer
  satisfaction, .
• For a different decision, it may be interested in
  reducing costs.

16
Alternatives and Criterion in Decision Making

• Alternatives are the choices available in a
  decision making situation.
• Sometimes, for emphasis, feasible alternatives is
  also used to rule out impractical alternatives.

17
Pricing Decision

• Demand 500 5 Price
• Criterion Maximize revenues.
• Revenue (500 5 P) P
• 500 P 5 P2
• Is this a linear function in price?

18
Pricing Decision

• Revenue function
• 0
  100
• Units

19
A Solution to Pricing Problem

• How do we identify the highest point of the
  revenue curve?
• Tangent to the total revenue curve has 0 slope
• Tangent of TR 500 P 5 P2
• Is 500 2 5 P 0
• Or P 50/unit

20
Classification of Management Science Models

• Deterministic Parameters of the problem are
  known.
• Stochastic or probabilistic Parameters are not
  known with certainty.
• Single Period
• Multiple Perod

21
Goal Seek

• Not attempting to find the best solution but
  satisfactory solution.
• Price 100, Variable cost 40
• Fixed costs 30,000 per year.
• How many units to sell to make annual profit of
  20,000?

22
Problems

• 7, 8, 9, 11, 14