Introduction To Modeling

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1-1

Introduction To Modeling

Chapter 1 Chapter Topics

 

The Management Science Approach to Problem Solving

 

Model Building: Break-Even Analysis

 

Computer Solution

 

Management Science Modeling Techniques

 

Business Usage of Management Science Techniques

 

Management Science Models in Decision Support Systems

The Management Science Approach

 

Management science uses a scientific approach to solving management problems.

 

It is used in a variety of organizations to solve many different types of problems.

 

It encompasses a logical mathematical approach to problem solving.

 

Management science, also known as operations research, quantitative methods, etc., involves a philosophy of problem solving in a logical manner.

Figure 1.1

The Management Science Process

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  Observation - Identification of a problem that exists (or may occur soon) in a system or organization.

  Definition of the Problem - problem must be clearly and consistently defined, showing its boundaries and interactions with the objectives of the organization.

  Model Construction - Development of the functional mathematical relationships that describe the decision variables, objective function and constraints of the problem.

  Model Solution - Models solved using management science techniques.

  Model Implementation - Actual use of the model or its solution.

Information and Data:

  Business firm makes and sells a steel product

  Product costs $5 to produce

  Product sells for $20

  Product requires 4 pounds of steel to make

  Firm has 100 pounds of steel Business Problem:

  Determine the number of units to produce to make the most profit, given the limited amount of steel available.

Example of Model Construction (1 of 3)

Variables: X = # units to produce (decision variable) Z = total profit (in $)

Model: Z = $20X - $5X (objective function) 4X = 100 lb of steel (resource constraint) Parameters: $20, $5, 4 lbs, 100 lbs (known values) Formal Specification of Model:

maximize Z = $20X - $5X subject to 4X = 100

Example of Model Construction (2 of 3) Example of Model Construction (3 of 3)

Solve the constraint equation:

4x = 100

(4x)/4 = (100)/4 x = 25 units

Substitute this value into the profit function:

Z = $20x - $5x

= (20)(25) – (5)(25) = $375

(Produce 25 units, to yield a profit of $375)

Model Solution:

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Model Building:

Break-Even Analysis (1 of 12)

■  Used to determine the number of units of a product to sell or produce that will equate total revenue with total cost.

■  The volume at which total revenue equals total cost is called the break-even point.

■  Profit at break-even point is zero.

Model Components

 

Volume (v) – the number of units produced or sold

 

Fixed Cost (c

_f

) - costs that remain constant regardless of number of units produced.

 

Variable Cost (c

_v

) - unit production cost of product.

 

Total variable cost (vc

_v

) - function of volume (v) and unit variable cost.

Model Building:

Break-Even Analysis (2 of 12)

Model Components

 

Total Cost (TC) - total fixed cost plus total variable cost.

 

Profit (Z) - difference between total revenue vp (p = unit price) and total cost, i.e.

Model Building:

Break-Even Analysis (3 of 12)

v

f

vc

c TC = −

v f

- vc vp - c Z =

Model Building:

Break-Even Analysis (4 of 12)

Computing the Break-Even Point

The break-even point is that volume at which total revenue equals total cost and profit is zero:

v f

f v

v f

c p v c

c c p v

vc c vp

= −

=

−

=

−

− ) (

0 The break-even point

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Break-Even Analysis (5 of 12)

Example: Western Clothing Company

Fixed Costs: c

_f

= $10,000 Variable Costs: c

_v

= $8 per pair Price : p = $23 per pair

The Break-Even Point is:

v = (10,000)/(23 -8) = 666.7 pairs

Break-Even Analysis (6 of 12) Figure 1.2

Model Building: Sensitivity Analysis Break-Even Analysis (7 of 12)

Example: Western Clothing Company

Fixed Costs: c

_f

= $10,000 Variable Costs: c

_v

= $8 per pair Price : p = $30 per pair (23)

The Break-Even Point is:

v = (10,000)/(30 -8) = 454.5 pairs

Model Building:

Break-Even Analysis (8 of 12) Figure 1.3

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Model Building:

Break-Even Analysis (9 of 12)

Example: Western Clothing Company

Fixed Costs: c

_f

= $10,000

Variable Costs: c

_v

= $12 per pair (8) Price : p = $30 per pair (23)

The Break-Even Point is:

v = (10,000)/(30 -12) = 555.5 pairs

Model Building:

Break-Even Analysis (10 of 12) Figure 1.4

Model Building:

Break-Even Analysis (11 of 12)

Example: Western Clothing Company

Fixed Costs: c

_f

= $13,000 (10,000) Variable Costs: c

_v

= $12 per pair (8) Price : p = $30 per pair (23)

The Break-Even Point is:

v = (13,000)/(30 -12) = 722.2 pairs

Model Building:

Break-Even Analysis (12 of 12) Figure 1.5

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Exhibit 1.1

Exhibit 1.2

Break-Even Analysis: Excel QM Solution (3 of 5) Break-Even Analysis: QM Solution (4 of 5)

Exhibit 1.4

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Break-Even Analysis: QM Solution (5 of 5)

Exhibit 1.5

Figure 1.6 Modeling Techniques

Classification of Management Science Techniques

 

Linear Mathematical Programming - clear objective;

restrictions on resources and requirements; parameters known with certainty. (Chap 2-6, 9)

 

Probabilistic Techniques - results contain uncertainty. (Chap 11-13)

 

Network Techniques - model often formulated as diagram;

deterministic or probabilistic. (Chap 7-8)

 

Other Techniques - variety of deterministic and probabilistic methods for specific types of problems including forecasting, inventory, simulation, multicriteria, etc. (Chap 10, 14-16)

Characteristics of Modeling Techniques

 

Some application areas:

- Project Planning - Capital Budgeting - Inventory Analysis - Production Planning

- Scheduling

 

Interfaces - Applications journal published by Institute for

Operations Research and Management Sciences (INFORMS)

Business Use of Management Science

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A decision support system is a computer-based system that helps decision makers address complex problems that cut across different parts of an organization and operations.

Features of Decision Support Systems

 

Interactive

 

Use databases & management science models

 

Address “what if” questions

 

Perform sensitivity analysis Examples include:

ERP – Enterprise Resource Planning OLAP – Online Analytical Processing

Figure 1.7 A Decision Support System

Introduction To Modeling

Introduction To Modeling

Chapter 1

Chapter Topics

The Management Science Approach to Problem Solving

Model Building: Break-Even Analysis

Computer Solution

Management Science Modeling Techniques

Business Usage of Management Science Techniques

Management Science Models in Decision Support Systems

The Management Science Approach

Management science uses a scientific approach to solving management problems.

It is used in a variety of organizations to solve many different types of problems.

It encompasses a logical mathematical approach to problem solving.

Management science, also known as operations research, quantitative methods, etc., involves a philosophy of problem solving in a logical manner.

Figure 1.1

The Management Science Process

 Observation - Identification of a problem that exists (or may occur soon) in a system or organization.

 Definition of the Problem - problem must be clearly and consistently defined, showing its boundaries and interactions with the objectives of the organization.

 Model Construction - Development of the functional mathematical relationships that describe the decision variables, objective function and constraints of the problem.

 Model Solution - Models solved using management science techniques.

 Model Implementation - Actual use of the model or its solution.

Information and Data:

 Business firm makes and sells a steel product

 Product costs $5 to produce

 Product sells for $20

 Product requires 4 pounds of steel to make

 Firm has 100 pounds of steel Business Problem:

 Determine the number of units to produce to make the most profit, given the limited amount of steel available.

Example of Model Construction (1 of 3)

Variables: X = # units to produce (decision variable) Z = total profit (in $)

Model: Z = $20X - $5X (objective function) 4X = 100 lb of steel (resource constraint) Parameters: $20, $5, 4 lbs, 100 lbs (known values) Formal Specification of Model:

maximize Z = $20X - $5X subject to 4X = 100

Example of Model Construction (2 of 3) Example of Model Construction (3 of 3)

Solve the constraint equation:

4x = 100

(4x)/4 = (100)/4 x = 25 units

Substitute this value into the profit function:

Z = $20x - $5x

= (20)(25) – (5)(25) = $375

(Produce 25 units, to yield a profit of $375)

Model Solution:

Model Building:

Break-Even Analysis (1 of 12)

■ Used to determine the number of units of a product to sell or produce that will equate total revenue with total cost.

■ The volume at which total revenue equals total cost is called the break-even point.

■ Profit at break-even point is zero.

Model Components

Volume (v) – the number of units produced or sold

Fixed Cost (c

) - costs that remain constant regardless of number of units produced.

Variable Cost (c

) - unit production cost of product.

Total variable cost (vc

) - function of volume (v) and unit variable cost.

Model Building:

Break-Even Analysis (2 of 12)

Model Components

Total Cost (TC) - total fixed cost plus total variable cost.

Profit (Z) - difference between total revenue vp (p = unit price) and total cost, i.e.

Model Building:

Break-Even Analysis (3 of 12)

vc

c TC = −

- vc vp - c Z =

Model Building:

Break-Even Analysis (4 of 12)

Computing the Break-Even Point

The break-even point is that volume at which total revenue equals total cost and profit is zero:

c p v c

c c p v

vc c vp

= −

=

−

=

−

− ) (

0

The break-even point

  Observation - Identification of a problem that exists (or may occur soon) in a system or organization.

  Definition of the Problem - problem must be clearly and consistently defined, showing its boundaries and interactions with the objectives of the organization.

  Model Construction - Development of the functional mathematical relationships that describe the decision variables, objective function and constraints of the problem.

  Model Solution - Models solved using management science techniques.

  Model Implementation - Actual use of the model or its solution.

  Business firm makes and sells a steel product

  Product costs $5 to produce

  Product sells for $20

  Product requires 4 pounds of steel to make

  Firm has 100 pounds of steel Business Problem:

  Determine the number of units to produce to make the most profit, given the limited amount of steel available.

■  Used to determine the number of units of a product to sell or produce that will equate total revenue with total cost.

■  The volume at which total revenue equals total cost is called the break-even point.

■  Profit at break-even point is zero.