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MBA 691 Introduction to Modeling and Linear

Programming

- Dr. Michael F. Gorman

Lecture Outline

- Introduction to the Analytic approach to business

problem solving - Analytical and experiential based decision making
- A framework for approaching business problem

solving - Introduction to Management Science/Operations

Research/ Decision Sciences/Production Operations

Management - Why study it?
- Introduction to modeling
- What is a model?
- What are attributes of a good model?
- What types of models will we cover in this class?
- Introduction to Mathematical Programming
- General structure of a math program
- Linear Programming
- Integer Programming
- Non-linear (quadratic) programming

Analytic Approach to Business Decision Making

- This course presents methods and techniques which

support - Quantitative
- Analytical
- Approach to business problem solving and decision

making - Quantitative
- Fact/data based support for decision making
- Analytical Approach
- Breaks down problem to its component parts to

allow better understanding of the behavior of the

component interactions and thus, the whole

problem

Experiential approach to business decision making

- Experiential approach
- Historical experience What happened last time?
- Expert opinion What will happen this time?
- Intuition what is the current feel for how

things work together? - Can analytical and Experiential approaches be

reconciled? - Trick is to balance the domain expertise of the

expert with the objectivity and thoroughness of

the analytic approach - The model is just another opinion at the table
- Although experiential and analytical approaches

often conflict, their strengths can be combined

to arrive at improved business decisions - Often the business expert (old head) and

technical (propeller head) expert are different

people, follow different career paths. - Move the process from combative to cooperative

Analytic vs. Experiential

- Experiential (Cons)
- Subjective bias, perception
- Limited Options Explored
- Can oversimplify/misstate component interactions
- Subject to organizational pressure
- Experiential (Pros)
- Can handle outliers well
- Handles non-quantifiables well

- Analytical (Pros)
- Objective
- Exhaustive/Thorough consideration of options
- Forces logical considerations of component

interactions - Analytical (Cons)
- Can misinterpret (or ignore) extreme information

to draw errant conclusions - Difficult to calibrate the non-quantifiables

We try to balance the insights analytics bring

to decision making from the insanity that can

result when methods are misapplied.

A framework for the decision making process

Observe/ Collect Data

Implement Evaluate

Problem Definition

Formulate Model

Verify Model

Select Alternative

Present Results

Feedback Loop

What is - Management Science - Operations

Research - Decision Science? What Operations

Research (Preferred term) Is In a nutshell,

operations research (O.R.) is the discipline of

applying advanced analytical methods to help make

better decisions. By using techniques such as

mathematical modeling to analyze complex

situations, operations research gives executives

the power to make more effective decisions and

build more productive systems based on - More

complete data - Consideration of all available

options - Careful predictions of outcomes and

estimates of risk - The latest decision tools

and techniques

In research, these are highly technical methods

for solving complicated problems (often in

business and engineering) In this class, these

topics are described as the use of quantitative

methods and analytical tools to solve business

problems.

Why study Management Science/Operations

Research/Decision Sciences?

- Build analytical thinking skills - focus on

relevant variables - Modeling/spreadsheet skills are valuable to

employers - for both producer and consumer of

model results - Leverage dominant technology - supply chain

planning/execution, yield mgmt, asset management - Exploit computer revolution - easier, cheaper

and more powerful tools - ubiquitous data what

to do with it? - Combine the MBA and the techie into one person -

eliminate the expert vs. analytical debate - Integrates other disciplines (marketing/finance/op

erations)

Introduction to Modeling

- A model is
- A simplified representation or approximation of a

real-world system - Designed to give insight into interrelationships

of key interrelating variables - Used to assist in drawing conclusions on the

real-world system - Examples of models
- Spreadsheet model revenue/cost/profit of a

business - Weather model prediction of precipitation,

temperature, wind - Engineering model structural integrity of a

physical structure - Sports model Sagarin team rankings
- Physiology life habits (diet/sleep/exercise)

and body response (weight/body fat/longevity) - Statistical relationship of random variables
- What is a good model? A good model is as simple

as possible, but no simpler. - A good model is useful model
- Provides insights into proper decisions
- Fits into business decision-making process
- A good model is a simple model
- Captures key attributes/relations of the real

world system, but no more

The role of assumptions

- An assumption removes some complexity of the

problem in order for it to be solvable. (AKA -

simplifying assumption) - Assumption of linear costs
- Assumption of no price impacts
- Assumption of no interaction between two

variables - Assumption of forecast accuracy
- Assumptions are your friends!
- Assumptions are necessary to solve problems
- Without them, we are stuck with real-world

complexity that is essentially unsolvable. - A good assumption
- Simplifies the problem statement/model

significantly - Is intuitively plausible to the domain experts
- Has negligible impact on the model results

Models covered in this class Prescriptive and

Descriptive

- Prescriptive models make a recommendation on a

course of action - Mathematical Programming models
- Also known as Optimization Models Linear

programming, Integer, quadratic - Given known inputs interaction and limitations
- Prescribe (or recommend) an optimal strategy
- Decision-making models
- Decision making under uncertainty (probabilistic

models) - Game-theoretic models (considering the reaction

of others to a decision) - Descriptive models describe and outcome of a

course of action - Statistical Models
- Specifically - Regression analysis
- Given a number of variables with unknown

quantitative relationship, - Estimate the impact of one set of variables on

another - Simulation Models
- Given random or unspecified variation in inputs

and dynamic interactions between inputs - Describe likely outcomes of a given set of inputs

(What-if analysis)

An example of a business problem which requires

multiple model types

- Optimal pricing for a suite of interrelated

products - Interrelated by their cost structures cost of

one affects the cost of another - Interrelated by their demand consumption of one

affects the consumption of another - Example Rail freight transportation services
- Regression analysis
- Estimates the cost curves, demand curves
- Optimization
- Recommends optimal prices, given costs and

demands - (may be a math program or decision-making/game

theoretic model) - Simulation
- Evaluates robustness of recommendations against

future unknown variations (e.g. customer response

to price change, shifts in demand)

Introduction to Math Programming

- A Math Programming Model has the following

ingredients - An Objective
- a mathematical expression
- to be maximized or minimized
- By changing values of decision variables
- E.g. Maximize Profits Minimize Total Distance,

etc. - And Constraints -
- Relationships among the decision variables which

somehow limits their use - Subject To used to identify the constraints
- E.g.
- Minimum output produced must be greater than or

equal to customer demand - Maximum used must be less than or equal to total

available inputs - Production of one product must equal production

of another

Types of Math programming

- Math programming has different functional

forms - Linear Program (LP)
- Objective function and constraints are linear
- This means decision variables are multiplied only

by a constant, never raised to a power or

multiplied times each other - Constraints are one of , , or
- All decision variables are continuous (can take

on a fractional value) - Integer Program (IP)
- Like an LP, but (some) decision variables can

take on only integer values - E.g. How many jobs are assigned to a machine
- Special case IP variables take on values of only

0 or 1 (binary) - E.g. Should a warehouse be open or not? (yes or

no nothing in between) - Quadratic Program (QP)
- Like an LP, but the objective can take on a

squared value - E.g. Pricing maximization, portfolio variance

minimization

An example Tables and Chairs

- A furniture manufacturer is deciding on tables

and chairs production for the upcoming quarter. - Each chair sold nets the manufacturer 20 Each

table makes 30 in profit - The manufacture has a supply of 500 board feet

each week and 100 labor hours to allocate - Each chair takes 10 board feet of wood each

table takes 20 board feet - Each chair requires 4 labor hours each table

takes 2 hours of labor - The manufacturer wants to produce no more than 40

chairs and no more than 20 tables - What should the manufacturer do?

Identifying the Key ingredients to the math model

- Decision Variables
- Manufacturer must decide how many tables and

chairs to produce - Call them T and C
- The objective and constraints must be functions

of these decision variables - Objective
- The manufacturers objective? Unstated in problem
- If it is to minimize costs should produce 0

units - If it is to maximize T or C, should produce T25

or C20 - It may be safe to assume (or clarify with the

business owner) that this manufacturer wants to

MAXIMIZE PROFITS - Maximize Profit 20C 30T
- Constraints
- Total board feet used 20T 10C 500
- Total labor used 4C 2 L 100
- Total Chairs C 40
- Total Tables T 20
- (Implied that T, C are both gt 0)

Algebraic Formulation Tables and Chairs Problem

- Objective
- Max Profit 20C 30T
- Subject to
- 4C 2T lt 100 (Labor)
- 10C 20T lt 500 (Wood)
- T lt20 (Tables)
- Clt 40 (Chairs)

Example in a Picture Tables or Chairs?

Table Constraint

Problem Maximize Profit Wood Supply 500 board

feet Labor Supply 100 hours Chair uses 10 wood

and 4 labor Table uses 20 wood and 2

labor Chair 20 profit Table 30

profit Customer orders 20 Tables, 40

chairs Problem Statement Max Profit 20C

30T Subject to 4C 2T lt 100 (Labor) 10C

20T lt 500 (Wood) T lt20 Clt 40

Chairs

50

Chair Constraint

40

Wood Constraint

25

(16.67T,16.67C)

(20T,10C)

Feasible Solution Space

Labor constraint

25

50

20

Tables

Linear Programming Model How does it work?

- The Simplex method
- Like knowing how internal combustion works in

order to drive a car - Nice to know, but not necessary
- Linear Programs often start with a feasible

solution - Meets all constraints
- Example produce 0 units (T0, C0)
- Model Looks at trade-offs between variables
- Benefit produced by increasing one output

(increased profit) - Compared to the cost of giving up an alternative

output (each chair means you give up some

capacity for building tables) - Change current solution by moving in

profit-improving directions - Continually try different combinations of outputs
- Until no profit-improving possibilities exist
- Efficiently checks only the intersections of

constraints - Because of the linear nature of the problem, we

know that the optimal solution will end up at a

corner of two constraints

Tables And Chairs Excel Formulation

Example in a Picture Tables or Chairs?

Table Constraint

Chairs

50

Chair Constraint

40

Wood Constraint

25

Optimum Solution Approx. (16.67T,16.67C)

S0 Profit 500

(20T,10C)

Labor constraint

25

S0 Profit0

20

S3 Profit 833

Tables

S2 Profit 800

S1 Profit 600

Adding Solver Add-In Tools Add Ins --- Solver

Add-in

If Solver Add-In is not Found in Tools Add Ins

You can find SOLVER.XLA in the directory

above. Browse to this directory, then

double-click to include.

Special Case LP Results that can occur

- Unbounded
- You want to maximize something, and there are no

constraints on the decision variables - The optimal solution is to produce infinite!!
- Unbounded problems are usually missing important

constraints, or that you have reversed a sign

that makes a bad thing good - Infeasible
- By the time you implement all the constraints,

the feasible region is null (nada, zip, zilch,

nothing, empty) - The problem has no solution
- Usually the constraints are implemented

incorrectly (units, etc.) - Sometimes a cost is represented as a constraint

Sensitivity Analysis

- Sometimes we want to understand how much an

optimum solution will change if some input data

changed - Remember, we have assumed perfect data and

information to this point - Examples
- How much would the Table constraint have to be

reduced before it became important has an

effect on the solution? - Currently, maximum tables is set at 20 if it

were set at 16.67 or less (a reduction of 3.33),

it would become important to (or change) the

solution - What if we could sell tables for a higher price

(more profit) Would that change the optimal mix? - This is like changing the slope of the isoprofit

curve - What would it be worth to expand labor or wood

constraints? - A parallel shift of one of the constraints

Sensitivity Analysis Making a loose table

constraint tight

Table Constraint

Chairs

50

Chair Constraint

40

Reducing the max table Constraint by 3.3 makes

it Tight further reductions reduces the

objective function as the optimal solution moves

along the labor constraint.

Wood Constraint

25

Optimum Solution Approx. (16.67T,16.67C)

Labor constraint

25

16.67

Tables

Sensitivity analysis Price of chairs falls by 5

Table Constraint

Wood constraint forces tradeoff Between tables

and chairs of 21 The profit ratio of tables and

chairs is 32 So we produce more chairs. If

tables fetched 10 more, or Chairs 5 less, then

we would Change the optimum to (20T, 10C)

Chairs

50

Chair Constraint

40

Wood Constraint

Optimum Solution Approx. (16.67T,16.67C) OR

(20T, 10C) Or anywhere in between

25

S0 Profit 375

(20T,10C)

Labor constraint

S0 Profit0

20

25

S3 Profit 750

Tables

S2 Profit 750

S1 Profit 600

Sensitivity analysis One more unit of labor

Table Constraint

Chairs

50

Chair Constraint

One more unit of labor allows 1/3 more chairs

1/6 less tables (as dictated by wood

constraint) (17C, 16.5 T) 6.67 in Chair

profit -5.00 in Table profit Net profit change

of 1.67

40

Wood Constraint

25

Optimum Solution Approx. (16.5T,17C)

Labor constraint

25

20

S3 Profit 835

Tables

Presentation of Results

- We recommend 16.67 chairs and 16.67 tables be

produced per week in order to earn 833 in profit

per week - All labor and wood inputs are fully utilized with

this solution - We suggest if it is possible, that we expand

labor as much as 70 hours per week in order to

gain 1.66 in profit per week increase per labor

hour (116 total potential)

Sensitivity analysis terms Adjustable Cells

- In Excel, two tables are produced in a

sensitivity analysis - Adjustable Cells and Constraints
- Adjustable Cells Sensitivity analysis
- Final Value optimal solution for the decision

variable - E.g. 16.67 Tables in our base solution
- Reduced cost amount the coefficient in the

objective function on a decision has to change

before it appears in the solution - No example in our problem both Tables and

Chairs have a positive value in the final

solution - Allowable increase/decrease amount an objective

function coefficient can change before changing

the optimal combination of outputs - E.g. Chair profit can go down 5, or Table profit

can go up 10, without changing the chair/table

mix of 16.67 apiece - Objective coefficient the input data value of

each decision variable to the objective function - E.g. Tables 30 Chairs 20

Sensitivity analysis terms Constraints

- Constraints
- Final value - the final value of the use of a

resource in a constraint - E.g. we use all 100 units of labor and 500 units

of wood in the base solution - Shadow Price the amount the objective function

would grow if the constraint were expanded by one

unit - E.g. The objective function increased by 1.67 by

expanding the labor constraint one unit - Allowable increase or decrease
- Amount a constraint can change before the shadow

price changes - In the case of the Table constraint, can decrease

the maximum table constraint by 3.33 before the

shadow price changes from 0 - Constraint R.H. (Right Hand) side
- Original value (input data) of the constraint

maximum or minimum - Called R.H. side because typically constraints

are stated as f(decision variables) lt max and

max is on the right hand side of the

inequality.

Problem with Chairs and Tables Solution!

- A marketing expert exasperatedly informs us that

it is crazy to produce equal numbers of chairs

and tables - First of all, we sell chairs at a ratio of no

less than 41 over tables - (that make sense)
- And chairs break more often than tables, so we

have a thriving replacement business - (We can exceed 41, but we better not fall below

it) - Going back to our operations expert, we inquire

if this is true. - Of course, he indicates, yes, its true,

why do you think I told you we never produce more

than 20 tables or 40 chairs?? - We now realize we never had a well-defined

problem so we go back and reformulate the

problem with this new constraint

Revised Table and Chairs Problem

Table Constraint

Chairs

50

Chair gt 4 Table

40

With the new 41 Chair to table constraint, more

chair must be produced. Labor is used up more

quickly, and we have excess wood.

Wood Constraint

New Optimum (5.56T,22.24C)

25

Feasible Solution Space

(16.67T,16.67C)

Labor constraint

25

50

Tables