Review Simulation, Inventory Models, Forecasting

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ReviewSimulation, Inventory Models, Forecasting

• Dr. A. Kleinstein, May 2005

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Simulation

• Simulation Using computer to replicate the
  characteristics of a real system.
• When the probability distributions of the input
  variable are known the computer can generate
  values that have those probability distributions.
  
• Discrete Distribution In Excel use rand() and
  the vlookup table. By hand, use a table of
  random numbers.
• The model may be expressed through a flow chart
  that shows what is done with the input variables.
• Program the model into the computer, or follow
  the model with hand calculations.
• Compute the relevant statistics.
• Examples Harrys Tire, Simpkins Hardware,
  Three Hill Power Company, Port of New Orleans,
  vlookup example, day 26.

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Linear Programming

• Linear Programming technique to maximize or
  minimize a linear function where the variables
  are subject to linear constraints. This means
  that the functions that represent the use of
  resources are linear functions, and that they are
  , or to constants. (The constants represent
  the total amount of resources available.)
• In applying this to real world problems the
  variables are the amount to produce, and we make
  the assumptions of
• certainty the coefficients of the objective and
  constraint functions are known and do not change
• proportionality the constraint equations
  accurately reflect the use of resources. Thus
  producing twice as much of a product uses twice
  the resources, etc.

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Linear Programming, cont

• additivity the objective function accurately
  reflects the objective. Thus, each product
  contributes additively to the objective function.
• divisibility the solution values that represent
  the amount of product to produce make sense even
  if they are fractional values.
• non-negativity all variables must be positive.
• When formulating a linear programming problem, be
  sure that the constraint equations are to the
  left of the equality of inequality signs, and the
  constants are to the right.

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Linear Programming, cont.

• Linear programming problems can be solved with
  Excel using the solver.
• When using the solver, put the values of the
  coefficients and the objective function in rows
  of cells, and starting values for the decision
  variables in a row of cells. Express the linear
  function using the Excel command sumproduct..
• The sensitivity report, or rerunning the solver
  with new values for the resources available, can
  tell how much the objective function will change
  for a unit change in resources available.
• Examples
• Chapter 7 Flair Furniture Company, Holiday Meal
  Turkey Ranch.
• Chapter 8 Media Selection, Marketing Research,
  Portfolio Selection, Greenberg Motors.
• Day 23 Portfolio Selection

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Inventory Models

• Inventory Models PowerPoint presentation, day
  16.
• Economic Order Quantity Model Used since 1915.
  Limited in application, but this model is useful
  as a conceptual tool, and starting off point.
• Assumption
• demand is known and constant
• lead time is known and constant
• receipt of inventory is instantaneous
• quantity discounts are not possible
• only variable costs are the cost of placing an
  order, the ordering cost, and the cost of holding
  or storing inventory over time, the holding or
  carrying cost.
• Orders are placed so that stockouts and shortages
  are completely avoided.

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Inventory Model, cont

• The minimum cost occurs when the carrying cost
  equals the holding cost. This gives the
  equations on page 197. EOQ is the amount to
  order each time, so as to minimize holding plus
  carrying cost.
• If we allow quantity discounts, then we must
  include the cost of purchasing the items.
• total cost material cost ordering cost
  carrying cost.
• In the case when quantity discounts are
  available, we often express the carrying cost as
  a percentage (I) of the unit cost (C).
• When quantity discounts are available, compute
  the EOQ amount for each available price. See
  quantity discount model steps on page 207.
• Example
• Chapter 6 Sumco Pump Company. Brass Department
  Store
• Day 18 Appack example and Excel solution

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Forecasting

• Forecasting PowerPoint presentation, day 9.
• Forecasting method
• Moving average
• weighted moving average
• linear regression trend projections and casual
  forecasting
• Measures of accuracy
• Graphs visual measure
• MAD
• MSE
• SE standard error of the estimate. When using
  linear regression for prediction, 95 of the time
  the actual value is expected to be within 2
  standard errors(2SE) of the predicted value.

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Forecasting, cont.

• Correlation coefficient when above 0.6 we
  generally accept that the regression line should
  be used for prediction.
• When using Excel, the slope is computed with the
  slope function, the y-intercept with the
  intercept function, and the correlation
  coefficient with the correl function. By hand,
  see formulas on page 156, 157, and 169
• Examples
• Chapter 5 Wallace Garden Supply, Midwestern
  Manufacturing Company, Triple A Construction
  Company
• Day 12, solved example.

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Review Models, Probability, Decision Theory

• Qant 305 Spring 2005
• Dr. A. Kleinstein

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Quantitative Analysis

• Mathematical tools have been used for thousands
  of years
• QA can be applied to a wide variety of problems
• One must understand the specific applicability
  of the technique, its limitations and its
  assumptions

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Input/Process/Output

• Scientific Approach to Managerial Decision Making
• Consider both Quantitative and Qualitative Factors

Meaningful Information
Quantitative Analysis
Raw Data
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The QA Approach
Define the Problem
Develop a Model
Acquire Input Data
Develop a Solution
Test the Solution
Analyze the Results
Implement the Results
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Modeling in the Real World

• Models are complex
• Models can be expensive
• Models can be difficult to sell
• Models are used in the real world by real
  organizations to solve real problems

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A Model Can be Mathematical Equations
Profit Model

• Profit Revenue - Expense

Revenue (Price per Unit) ? (Quantity Sold)
Expenses Fixed Cost - (Variable Cost/Unit)
? (Quantity Sold)
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Breakeven Quantity Model
P price Q quantity sold F fixed cost V
variable cost/unit
Profit PQ-F-VQ
Solve for Q, the breakeven quantity by setting
profit 0 F PQ VQ, thus Q
F/(P V)
Breakeven Quantity F/(P-V)
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Mathematical Models Characterized by Risk

• Deterministic models - we know all values used in
  the model with certainty
• Probabilistic models - we know the probability
  that parameters in the model will take on a
  specific value

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Probability

• Life is uncertain!
• We must deal with risk!
• A probability is a numerical statement about the
  likelihood that an event will occur
• 0 ? P(event) ? 1

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Example Rolling a 6-sided Die

• Outcome
• of Roll
• 1
• 2
• 3
• 4
• 5
• 6

• Probability
• 1/6
• 1/6
• 1/6
• 1/6
• 1/6
• 1/6
• Total 1

Rolling a die has six possible outcomes
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Random Variables

• Discrete random variable - can assume only a
  finite or limited set of numeric values- i.e.,
  the number of automobiles sold in a year.
• Continuous random variable - can assume any one
  of an infinite set of numeric values - i.e.,
  temperature, product lifetime.

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Discrete Random Variable
A random variable includes the possible outcomes
and the probability of getting those outcomes.
Probability Distribution
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Expected Value of a Discrete Random Variable
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Variance of a Discrete Random Variable
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Continuous Random Variable

• Normal distribution

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Area Under the Curve gives Probability
P(X 26
Areas Under the Normal Curve
One, two, and three standard deviations from the
mean.
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Excel Functions for aNormal Random Variable

• Normdist(X,µ,s,true) P(x
• Norminv(prob, µ, s) X, where P(x

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Area Under the Normal Curve using Excel, Example 1
P(X ,true)
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Area Example 2
P(X rue)
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Area Example 3
P(110 Normdist(125,100,20,true) Normdist(110,100,20
,true)
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Decision Making Under Risk

• Risk means that you know the probability of
  occurrence for each state of nature.

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Decision-Making Under Risk

• Clearly define the problem at hand
• List the possible alternatives
• Identify the possible outcomes (states of nature)
  and probabilities of getting those outcomes.
• List the payoff or profit of each combination of
  alternatives and states of nature.
• Compute the Expected Monetary Value (EMV), and
  choose the alternative with the highest EMV.

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Decision-Making Under Risk
Choose the alternative with the highest Expected
Monetary Value (EMV)
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Example Three alternatives and Two States of
Nature
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Choose Alternative with the Highest EMV
Choose this alternative. Highest EMV.
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Perfect Information

• Perfect information means that you know the state
  of nature that will occur.
• You might know this because of market research.
• Knowing the state of nature does NOT alter the
  probability that the state will occur. It only
  means that you have determined which of the
  states will occur before you choose the
  alternative.

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Expected Value With Perfect Information (EV PI)
Example EV PI 200,000.50 0.50 100,000
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Expected Value of Perfect Information (EVPI)

• EVPI places an upper bound on what one would pay
  for additional information
• EVPI is the expected value with perfect
  information minus the maximum EMV

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Calculating EVPI

• EVPI EV PI - max(EMV)
• Example
• EVPI 100,000 - 40,000 60,000

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Expected Opportunity Loss

• EOL is the cost of not picking the best solution
• EOL Expected Regret
• EOL best payoff in a column payoff in column
  cell.

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Computing EOL - The Opportunity Loss Table
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Making a Decision with EOL Table Under Risk

• Calculate the EMV for each alternative in the EOL
  table.
• Choose the alternative with the minimum EMV.
• The value of the minimum EMV is also the EVPI.