Introduction to Management Science Chapter 1: Hillier and Hillier

Agenda  Definition of Management Science  The Science of Management Science  Example of Spreadsheet Modeling/Analysis

Management Science  Hillier and Hillier define it as “a discipline that attempts to aid managerial decision making by applying a scientific approach to managerial problems that involve quantitative factors.” –Traditionally management science was called operations research.

Keys to Management Science  It is a discipline –There is a body of knowledge that is built upon a scientific foundation. –There are professional societies that meet about management science topics. Institute for Operations Research and the Management Sciences (INFORMS) International Federation of Operational Research Societies (IFORS)

Keys to Management Science Cont.  Aids managerial decision making –It is meant to provide an analysis and recommendations to solving a problem based on quantitative factors. –Management science is not well suited for examining intangible issues. Management needs to incorporate the quantitative results from a management science study with the intangible considerations. –Management science can flush out solutions that management may not have been able to fathom.

Keys to Management Science Cont.  Uses a scientific approach –Step 1: Define the problem and gather data –Step 2: Formulate a model (typically a mathematical model) to represent the problem –Step 3: Develop a computer-based procedure for deriving solutions to the problem from the model. –Step 4: Test the model and refine it as needed. –Step 5: Apply the model to analyze the problem and develop recommendations for management. –Step 6: Help to implement the team’s recommendations that are adopted by management.

Defining the Problem and Gathering Data  Usually, the first task in a management science study is to take a vague and imprecisely defined problem and hone it to a well-defined statement. –This includes identifying the objectives and constraints.  The second task is to gather the relevant data for the study. –This task can take a considerable amount of time, especially when the needed data is nonexistent.

Formulating a Mathematical Model  A mathematical model is an approximate representation of the problem that captures the key components of the problem and usually leaves out factors that have negligible effects. –Since a mathematical model is an abstraction of the real problem, there is usually no correct model, only better models.

Formulating a Mathematical Model Cont.  The key components of a mathematical model are: –Assumptions –Decision variables –Objective function –Constraints –Parameters/technical coefficients

Assumptions  An assumption is a fact or statement taken for granted.  Every mathematical model is built upon some group of assumptions.  Assumptions are usually needed to keep the problem tractable.

Decision Variables  These variables are ones that can change in the model in order to optimize the objective function while meeting the various constraints.  These variables are the ones that management has the ability to change.  Mathematical representation of n decision variables: x 1, x 2, …, x n

Objective Function  It is a function that describes an appropriate measure of performance.  Examples include profit functions and cost functions.

Constraints  These are the restrictions that affect the quantities that the decision variables can take.  They define the feasible set of solutions.  Examples of these include water constraints, labor constraints, and capital constraints.

Constraints Cont.  Types of constraints –Functional constraints These constraints will take-on a functional form, e.g., linear, quadratic, logarithmic, etc. Functional constraints can be either binding, i.e., met with equality, or non-binding, i.e., met with strict inequality. –Non-negativity constraints These constraints restrict a decision variable to either being positive or equal to zero.

Example of Constraints

Parameters/Technical Coefficients  These coefficients are usually constants that define what the impact would be on either the objective function or one of the constraints of changing a decision variable. –Example: 5x 1 + 7x 2 where the numbers 5 and 7 are the parameters.

Example of a Full Mathematical Model

Example 2 of a Full Mathematical Model

Develop a Computer-Based Procedure for Deriving Solutions  In this step the management science team chooses the best software that will solve the problem. –If the current available software is not adequate, the team may need to build its own.  It is crucial to have an understanding of the different solution algorithms that exist to be able to solve the problem efficiently.  If the mathematical model is well developed, determining an efficient algorithm should be straightforward.

Test the Model and Refine It As Needed  When working with a large problem with many decision variables and constraints, there will be bugs and flaws in the model that will need to be worked out.  In small problems it is easy to check if the model is being simulated correctly through graphical analysis, while with larger models you must depend on intuition.

Test the Model and Refine It As Needed Cont.  During this step it is useful to start with a smaller simplified model to begin with and then building it up in steps.  Testing and improving a model to increase its validity is commonly referred to as model validation.

Questions to Ask During Model Validation  Have all the relevant factors and interrelationships in the problem been accurately incorporated into the model?  Does the model seem to provide a reasonable solution?  When assumptions about costs and revenue are changed, do the solutions change in a plausible manner?

Apply the Model to the Problem and Develop Recommendations  Once an acceptable model has been developed the next step is to apply it to the problem at hand and develop recommendations.  Usually more than one iteration of the model is run varying the assumptions.  If this model is meant to be used on a continual basis, documentation needs to be developed so anyone can understand the model and the computer program used to solve it.

Implement Recommendations Adopted by Management  Since change in many organizations is difficult, you may be required as the model builder to provide justification of why this solution will be helpful to the organization. –This may include explaining and justifying the assumptions you made in the model.  In some cases you may be required to assist in implementing the system that may be used on a continual basis.

Useful Excel Function: Name  This function in Excel allows you to name a cell or range of cells. –To Name a cell(s): Click on the cell or cells you would like to name Choose Insert at the top of the menu Choose Name and then Define Next type in a name in the dialog box  If(a,b,c)  Min(a,b) or Max(a,b)

Useful Excel Function: If(a,b,c)  The If(a,b,c) statement allows you to put a conditional statement in Excel. –This statement reads that if condition a is true, then do b, otherwise do c. –An if statement can have up to 7 nested if statements within it.

Useful Excel Function: Max() and Min()  The Max() returns the maximum number from an array of numbers you select.  The Min() returns the minimum number from an array of numbers you select.

Example 1 of Spreadsheet Analysis  Suppose you are a producer of dry beans. –Your sunk cost is $50,000 if you choose to produce. –Your variable cost is $650 per acre –Your revenue per acre is $700 per acre –What is your long-run breakeven quantity (BEQ)? Note: BEQ=Fixed cost/(revenue - variable cost)

Example 2 of Spreadsheet Analysis  Suppose you produce grandfather clocks.  You have a fixed cost of $50,000 if you produce, otherwise it is zero.  Your variable cost to produce a clock is $400.  Your revenue is $900 for producing a clock  What is the profit at Q=0, Q=50, Q=200 and what is the breakeven quantity?

Example 2 of Spreadsheet Analysis Cont.  The Model –Define Q as the number of clocks produced where Q  0. –Profit = 0 if Q = 0, otherwise –Profit = revenue*Q-variable cost*Q-fixed cost –BEQ=Fixed cost/(revenue - variable cost)  Solution will be done in class.

Example 2 Revisited  Now suppose the maker of grandfather clocks has a limit of s on the amount that he can sell.  When s =300, what is the profit at Q=0, Q=50, Q=200, Q=300, Q=400 and what is the breakeven quantity?  Is there a problem with how Hillier and Hiller calculate breakeven quantity?

Example 2 Revisited Cont.  The Model –Define Q as the number of clocks produced where Q  0 and s  Q. –Profit = 0 if Q = 0, otherwise –Profit = revenue*Q-variable cost*Q-fixed cost –BEQ=Fixed cost/(revenue - variable cost)  Solution will be done in class.