1. Managerial Decision Making and Problem Solving
Lecture Notes 1

2. Introduction
The body of knowledge involving quantitative approaches to decision making is referred to as
Management Science
Operations Research
Decision Science

3. What is Management Science?
Management Science is the discipline that adapts the scientific approach for problem solving to help managers make informed decisions.
The goal of management science is to recommend the course of action that is expected to yield the best outcome with what is available.

4. What is Management Science?
The basic steps in the management science problem solving process involves
Analyzing business situations (problem definition);
Building mathematical models to describe the business situation(Mathematical Modeling);
Solving the mathematical models;
Communicating/implementing recommendations based on the models and their solutions.

5. Problem Definition Steps to be taken Observe operations
Ease into complexity
Recognize political realities
Decide (identify) what is really wanted
Identify constraints
Seek continuous feedback

6. Mathematical Modeling
This is a procedure that recognizes and verbalizes a problem and then quantifies it using mathematical expressions.
Steps to be taken
Identify decision variables
Quantify the objective and constraints
Construct a model shell
Data gathering – consider time/cost issues

7. Classification of Mathematical Models
Classification by the model purpose
Optimization models
Prediction models
Classification by the degree of certainty of the data in the model
Deterministic models
Probabilistic (stochastic) models

8. Solving the Mathematical Model
Steps to be taken
Choose an appropriate technique
Generate model solution
Test/ validate model results
Return to modeling step if results are unacceptable
Perform “what-if” analysis

9. Communication / Implementation
Prepare a business report
Be concise
Use everyday language
Make liberal use of graphics
Monitor the progress of the implementation

10. Communication / Implementation
Components of a business presentation
Introduction
Assumptions/approximations made
Solution approach/ computer program used
Results
What-if analysis
Overall recommendation
Appendices

11. Management Science Applications
Linear Programming was used by Burger King to find how to best blend cuts of meat to minimize costs.
Integer Linear Programming model was used by American Air Lines to determine an optimal flight schedule.
The Shortest Route Algorithm was implemented by the Sony Corporation to developed an onboard car navigation system.

12. Management Science Applications
Project Scheduling Techniques were used by a contractor to rebuild Interstate 10 damaged in the 1994 earthquake in the Los Angeles area.
Decision Analysis approach was the basis for the development of a comprehensive framework for planning environmental policy in Finland.
Queuing models are incorporated into the overall design plans for Disneyland and Disney World, which lead to the development of ‘waiting line entertainment’ in order to improve customer satisfaction.

13. Introduction to Linear Programming
A Linear Programming model seeks to maximize or minimize a linear function, subject to a set of linear constraints.
The linear model consists of the following components:
A set of decision variables.
An objective function.
A set of constraints.

14. Introduction to Linear Programming
A feasible solution satisfies all the problem's constraints.
An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing).
A graphical solution method can be used to solve a linear program with two variables.

15. Introduction to Linear Programming
There are efficient solution techniques that solve linear programming models.
The output generated from linear programming packages provides useful “what if” analysis.

16. Introduction to Linear Programming
The Importance of Linear Programming
Many real world problems lend themselves to linear programming modeling.
Many real world problems can be approximated by linear models.
There are well-known successful applications in:
Manufacturing
Marketing
Finance (investment)
Advertising
Agriculture

17. Introduction to Linear Programming
Assumptions of the linear programming model
The parameter values are known with certainty.
The objective function and constraints exhibit constant returns to scale.
There are no interactions between the decision variables.
The Continuity assumption: Variables can take on any value within a given feasible range.

18. Example 1
Max 5x1 + 7x2 s.t. x1 < 6 2x1 + 3x2 < 19 x1 + x2 < 8 x1 > 0 and x2 > 0

19. Graphical Analysis – the Feasible RegionX2
The non-negativity constraints X1

20. Example 1: Graphical Solution
x1 + x2 = 8 8 7 6 5 4 3 2 1
x1 = 6
2x1 + 3x2 = 19
Feasible
Region x1

21. The search for an optimal solution
Start at some arbitrary value for objective function, then increase the value of the objective function, if possible, and continue until it becomes infeasible.

22. x2 8 7 6 5 4 3 2 1
5x1 + 7x2 = 35
5x1 + 7x2 = 39
5x1 + 7x2 = 42 x1

23. Example 1: Graphical Solution
Maximum
Objective Function Line
5x1 + 7x2 = 46 8 7 6 5 4 3 2 1
Optimal Solution
(x1 = 5, x2 = 3) x1

24. Extreme points and optimal solutions
There are three types of feasible points
Interior points
Boundary points
Extreme points
If a linear programming problem has an optimal solution, an extreme point is optimal.

25. Example 1: Extreme Points8 7 6 5 4 3 2 1 5
(0, 6 1/3) 4
(5, 3)
Feasible
Region 3
(6, 2) 1
(0, 0) 2
(6, 0) x1

26. Multiple optimal solutions
For multiple optimal solutions to exist, the objective function must be parallel to one of the constraints.

27. The Role of Sensitivity Analysis of the Optimal Solution
Is the optimal solution sensitive to changes in input parameters?
Possible reasons for asking this question:
Parameter values used were only best estimates.
Dynamic environment may cause changes.
“What-if” analysis may provide economical and operational information.

28. Slack and Surplus Variables
A linear program in which all the variables are non-negative and all the constraints are equalities is said to be in standard form.
Standard form is attained by adding slack variables to "less than or equal to" constraints, and by subtracting surplus variables from "greater than or equal to" constraints.
Slack and surplus variables represent the difference between the left and right sides of the constraints.
Slack and surplus variables have objective function coefficients equal to 0.