1. Introduction BSAD 30 Dave Novak
Source: Anderson et al., 2013 Quantitative Methods for Business 12th edition – some slides are directly from J. Loucks © 2013 Cengage Learning

2. Overview Background Problem solving and decision making
Quantitative analysis
Models of cost, revenue, and profit
Iron Works example
Ponderosa example

3. Background Management science Operations research Decision science
Business analytics
These terms are often used interchangeably
Originated during WWII – military strategic and tactical problems (choosing supply routes, storing material, getting the proper supplies from one point to another, etc.)

4. Background Simplex method for solving linear programming problems
Modern computing power and lots of data have resulted in all kinds of applications
Airline industry
Production and manufacturing
Vehicle routing (mail, delivery, etc.)

5. Problem solving and decision making
7 Steps of Problem Solving
(First 5 are the process of decision making)
1. Identify and define the problem
2. Determine the set of alternative solutions
3. Determine the criteria for evaluating alternatives
4. Evaluate the alternatives
5. Choose an alternative (make a decision)
6. Implement the selected alternative
7. Evaluate the results

6. Problem solving and decision making
This seems trivial and intuitive – why do we even need to discuss this?

7. Quantitative analysis and decision making
Further breaking down the decision-making process
Problems with single objective (one criterion) are single-criterion
Problems with multiple objectives (more than one criteria) are multicriteria

8. Quantitative analysis and decision making
Focus on being able to numerically measure the data associated with the problem
Develop mathematical expressions that describe objectives, constraints and other relationships in the problem
Use some type of quantitative methods approach to make a recommendation

9. Quantitative analysis and decision making
Quantitative analysis process (4 steps)
Model development
Data preparation
Model solution
Generate output or reports

10. Model development A model is some type of representation of reality
There are three (3) forms of models:
Iconic models – physical replicas of real objects (i.e. model car, a model of a building, etc.)
Analog models – physical form, but do not physically represent object being modeled (speedometer, thermometer, graph of social network, etc.)

11. Model development
Mathematical models – using mathematical formulas and expressions to represent a real world problem (total profit, total cost, etc.)
Profit (P) = 10x, where x is the number of units sold and $10 is the profit from each unit

12. Some different models Some types of models Maps (2 dimensions)
Music scores
Architectural drawings
Data flow diagrams
Mathematical models
Max P = 18x1 + 12x2
Subject to
0.16x x2 ≤ 0 (Cutting)
0.47x x2 ≤ 0 (Sewing)
0.40x x2 ≤ 0 (Decorating)
x1, x2 ≥ 0 (Non negativity)

13. Why use models? Experimenting with models generally:
requires less time
is less expensive
involves less risk
can enable you to investigate a situation that cannot be represented in reality
The more closely the model represents the real situation, the more accurate the conclusions and predictions will be

14. Why use models? Models are attempts to represent reality
“Essentially, all models are wrong, but some are useful” quote attributed to statistician George Box
In practice, models rarely capture “the exact” or “full” reality of a given situation

15. Mathematical models
Objective Function – a mathematical expression that describes the problem’s objective, such as maximizing profit or minimizing cost
P=10x (from slide #11)
10x is the OBJECTIVE FUNCTION

16. Mathematical models
Constraints – a set of restrictions, limitations, or assumptions such as a limit in production capacity or a fixed number of labor hours
Take our production problem P=10x (from slide #11)
Assume 5 hours of labor are required to produce each unit of x, and the total hours of labor available each week are 40
5x ≤ 40 is the LABOR CONSTRAINT

17. Mathematical models
Uncontrollable Inputs – environmental factors that are not under the control of the decision maker
In our simple model, the profit per unit ($10), the production time per unit (5 hours), and the production capacity (40 hours) are environmental factors not under the control of the manager or decision maker

18. Mathematical models
Decision Variables – controllable inputs; choices made by the decision maker, such as the number of units of a product to produce
In our model, the decision maker (or manager) has one choice – how much of x to produce

19. Mathematical models
Our complete mathematical model for the simple production problems is:
Maximize Profit 10x (objective function)
Subject to: ) 5x ≤ 40 (labor constraint)
2) x ≥ 0 (non-negativity constraint)

20. Mathematical models Mathematical models can be:
Deterministic - if all uncontrollable inputs (profit, labor hours, etc.) to the model are known with certainty and cannot vary
For this class, we commonly make this assumption, although in reality this is RARELY the case

21. Mathematical models
Stochastic (or probabilistic)- if any uncontrollable inputs (profit, labor hours, etc.) are uncertain and subject to variation
For example, if the number of hours of production time per unit of x could vary between 3 and 6 hours – right now, we assume this value is fixed at 5
Obviously the “answer” to the problem (how many units of x should we produce?) will change if the production time changes

22. Mathematical models
Cost/benefit, usability, whether managers understand the model, etc. – these are considerations that must be made when selecting a model
It could be that a less complicated and less precise model is more appropriate than a more complicated model due to cost and ease of use considerations

23. Transforming inputs to outputs
$10 profit per unit
5 labor hours per unit
40 – maximum labor hours / week
Uncontrollable Inputs
(Environmental Factors)
Our value for production quantity (say x=8)
Profit = 80
Labor hours = 40
Controllable
Inputs
(Decision
Variables)
Output
(Projected
Results)
Mathematical
Model
Max P: 10(8)
s.t (8) ≤ 40
8 ≥ 0

24. Data preparation Data preparation is not a trivial step in modeling
Time
Data errors
Understanding of data
A model with 50 decision variables (x1, x2, … x50) and 25 constraints could have over 1,300 data elements
Might need IS expertise

25. Model solution
The analyst attempts to identify the alternative (the decision variable values) that provides the “best” output for the model
The “best” output is the optimal solution
If the alternative does not satisfy all of the model constraints, it is rejected as being infeasible, regardless of the objective function value
If the alternative satisfies all of the model constraints, it is feasible and a candidate for the “best” solution

26. Model solution
On page 12, the book illustrates a trail-and-error solution
Production
Projected
Total Hours
Feasible
Quantity
Profit
of Production
Solution
Yes 2 20 10 4 40 6 60 30 8 80
100 50 No 12
120

27. Model testing and validation
Goodness/accuracy of a model may not be assessed until solutions are generated
Small test problems having known, or at least expected, solutions can be used for model testing and validation
If the “test” model generates expected solutions, use the model on the full-scale problem

28. Model testing and validation
If inaccuracies or potential shortcomings inherent in the model are identified, take corrective action such as:
Collection of more-accurate input data
Modification of the model

29. Reports / output
A managerial report, based on the results of the model, should be prepared
Report should be easily understood by the decision maker and should contain the analyst’s interpretation of the results
recommended decision
other pertinent information about the results (for example, how sensitive the model solution is to the assumptions and data used in the model)

30. Implementation and follow-up
Successful implementation of model results is of critical importance
Secure as much user involvement as possible throughout the modeling process
Continue to monitor the contribution of the model
It might be necessary to refine or expand the model
Things change over time, and so should most models

31. Iron Works, Inc. example
Iron Works, Inc. manufactures two products made from steel and just received this month's allocation of b pounds of steel. It takes a1 pounds of steel to make a unit of product 1 and a2 pounds of steel to make a unit of product 2.
Let x1 and x2 denote this month's production level of
product 1 and product 2, respectively
Let p1 and p2 denote the unit profits for products 1 and 2, respectively
Iron Works has a contract calling for at least m units of product 1 this month. The firm's facilities are such that at most u units of product 2 may be produced monthly

32. Iron Works, Inc. example Write the mathematical model
What is our objective function verbally?
What is our objective function mathematically?

33. Iron Works, Inc. example Write the mathematical model
What are our constraints verbally?

34. Iron Works, Inc. example Write the mathematical model
What are our constraints mathematically?

35. Iron Works, Inc. example
Write the mathematical model
Full problem

36. Iron Works, Inc. example Solve for a particular situation
Suppose b = 2000, a1 = 2, a2 = 3, m = 60, u = 720, p1 = 100, and p2 = 200
Rewrite the model with these specific values for the uncontrollable inputs

37. Iron Works, Inc. example Uncontrollable Inputs
$100 profit per unit Prod. 1
$200 profit per unit Prod. 2
2 lbs. steel per unit Prod. 1
3 lbs. Steel per unit Prod. 2
2000 lbs. steel allocated
60 units minimum Prod. 1
720 units maximum Prod. 2
0 units minimum Prod. 2
60 units Prod. 1
units Prod. 2
Max 100(60) + 200(626.67)
s.t. 2(60) + 3(626.67) < 2000
> 60
< 720
>
Profit = $131,333.33
Steel Used = 2000
Controllable Inputs
Output
Mathematical Model

38. Iron Works, Inc. example
What is a potential issue with the solution for the example problem?

39. Ponderosa Development Corp. example
Ponderosa Development Corporation (PDC) is a small real estate developer that builds only one style cottage. The selling price of the cottage is $115,000
Land for each cottage costs $55,000 and lumber, supplies, and other materials run another $28,000 per cottage
Total labor costs are approximately $20,000 per cottage

40. Ponderosa Development Corp. example
Ponderosa leases office space for $2,000 per month
The cost of supplies, utilities, and leased equipment runs another $3,000 per month
The one salesperson of PDC is paid a commission of $2,000 on the sale of each cottage
PDC has seven permanent office employees whose monthly salaries are given on the next slide

41. Ponderosa Development Corp. example
Employee Monthly Salary President $10,000 VP, Development 6,000 VP, Marketing 4,500 Project Manager 5,500 Controller 4,000 Office Manager 3,000 Receptionist 2,000

42. Ponderosa Development Corp. example
Identify all the fixed costs components and the marginal cost and revenue for each cottage

43. Ponderosa Development Corp. example
Write out the monthly cost function
Write out the monthly revenue function
Write out the monthly profit function

44. Ponderosa Development Corp. example
What is the breakeven point for monthly cottage sales?

45. Ponderosa Development Corp. example
What is the monthly profit if 12 cottages per month are built and sold?

46. Ponderosa Development Corp. example
1200
Total Revenue =
115,000x
1000
800
Thousands of Dollars
600
Total Cost =
40, ,000x
400
200
Break-Even Point = 4 Cottages 1 2 3 4 5 6 7 8 9 10
Number of Cottages Sold (x)

47. Summary Background Problem solving and decision making
Quantitative analysis
Different types of models
How to set up a mathematical model
Models of cost, revenue, and profit
Iron Works example
Ponderosa example