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MBA 691 Introduction to Modeling and Linear Programming

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Title: MBA 691 Introduction to Modeling and Linear Programming


1
MBA 691 Introduction to Modeling and Linear
Programming
  • Dr. Michael F. Gorman

2
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

3
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

4
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

5
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.
6
A framework for the decision making process
Observe/ Collect Data
Implement Evaluate
Problem Definition
Formulate Model
Verify Model
Select Alternative
Present Results
Feedback Loop
7
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
8
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.
9
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)

10
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

11
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

12
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)

13
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)

14
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

15
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

16
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?

17
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)

18
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)

19
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
20
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

21
Tables And Chairs Excel Formulation
22
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
23
Adding Solver Add-In Tools Add Ins --- Solver
Add-in
24
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.
25
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

26
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

27
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
28
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
29
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
30
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)

31
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

32
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.

33
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

34
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
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