MBA 691 Introduction to Modeling and Linear Programming

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MBA 691 Introduction to Modeling and Linear
Programming

• Dr. Michael F. Gorman

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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

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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

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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

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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.
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A framework for the decision making process
Observe/ Collect Data
Implement Evaluate
Problem Definition
Formulate Model
Verify Model
Select Alternative
Present Results
Feedback Loop
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What is - Management Science - Operations
Research - Decision Science?What Operations
Research (Preferred term) IsIn 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
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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.
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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)

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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

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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

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Models covered in this classPrescriptive 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)

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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)

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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

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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

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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?

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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)

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Algebraic FormulationTables 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)

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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
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Chair Constraint
40
Wood Constraint
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(16.67T,16.67C)
(20T,10C)
Feasible Solution Space
Labor constraint
25
50
20
Tables
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Linear Programming ModelHow 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

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Tables And Chairs Excel Formulation
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Example in a Picture Tables or Chairs?
Table Constraint
Chairs
50
Chair Constraint
40
Wood Constraint
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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
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Adding Solver Add-InTools Add Ins --- Solver
Add-in
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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.
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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

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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

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Sensitivity AnalysisMaking 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
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Optimum Solution Approx. (16.67T,16.67C)
Labor constraint
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16.67
Tables
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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
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S0 Profit 375
(20T,10C)
Labor constraint
S0 Profit0
20
25
S3 Profit 750
Tables
S2 Profit 750
S1 Profit 600
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Sensitivity analysisOne more unit of labor
Table Constraint
Chairs
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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
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Wood Constraint
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Optimum Solution Approx. (16.5T,17C)
Labor constraint
25
20
S3 Profit 835
Tables
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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)

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Sensitivity analysis termsAdjustable 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

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Sensitivity analysis termsConstraints

• 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 asf(decision variables) lt maxand
  max is on the right hand side of the
  inequality.

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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

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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)
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Feasible Solution Space
(16.67T,16.67C)
Labor constraint
25
50
Tables