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## PowerPoint Slideshow about 'Simulation' - brinda

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Presentation Transcript

Presentation Outline

- Introduction to Simulation
- Tools of the trade:
- Tables, RAND, VLOOKUP, TABLES, NORMDIST, STDEV, Confidence Intervals
- Simulation of decision trees
- Simulation of regression curves
- Simulation of operations

Why Simulation?

- Quick and easy to use.
- Versatile enough to model any system.
- Captures system dynamics and variation.
- Shows system behavior over time.
- Force one to think through the operational details of a system.

Spreadsheets

- Spreadsheets are usually used to capture numerous, complex mathematical relations
- Very effective for communicating numerical models
- However, typically they do not capture dynamically changing interactions
- they offer static relations
- Usually, randomness and unpredictability within the system are not represented
- often, inadequate “scenarios” are presented
- Often, averages or end-states are used which can misrepresent interim states of the system

Simulation maximizes performance in the least amount of time

Potential

Simulation

System

Performance

Trial and error

SystemLife

Time

Variability

- Uncertainty, randomness, unpredictability, fluctuations all exist and create variability
- Variability has uncertain impact on system performance
- Especially when considered in tandem with interdependencies - variability destroys predictability.
- Why not get rid of variability?
- We just can’t.
- Deming: always better to reduce sources of variability where ever possible (including our own reactions to inherent variability).

Simulation in Systems Management

- Production/Customer Scheduling
- Resource Scheduling
- Maintenance Scheduling
- Work Prioritization
- Flow Management
- Delay/Inventory management
- Quality Management

Reason for Increased Use of Simulation

- Increased awareness of the technology.
- Increased availability, capability and ease-of-use of simulation software.
- Increased computer memory and processing speeds (especially at the PC level).
- Declining computer and software costs.

When to Simulate

- If the system is stochastic -- at least one random variable, even the statistical average results are never 100% certain.
- If system has interdependencies and variability.
- If interested in observing behavior over time.
- If the importance of the decision warrants the time and effort of a simulation.

Typical Random Variables

- Durations -- Operation times, repair times, setup times, move times
- Outcomes -- The success of an operation, the decision of which activity to do next
- Quantities -- Lot sizes, arrival quantities, number of workers absent.
- Eventintervals -- Time between arrivals, time between equipment failures
- Attributes -- Customer type, part size, skill level

Tools of the trade

- We will use Excel in this class – it is everywhere
- As a general-purpose tool, it is useful, but not nicely customized to simulation
- Decision Tree Simulation:
- Use @Risk (or Crystal Ball) add-on to Excel
- Operations Simulation:
- Use Promodel or similar animated simulation software

Excel Tools

- RAND() Generates uniform random variables between 0 and 1
- NORMDIST uses RAND seed to generate normal random variables
- VLOOKUP()
- Database function in Excel to lookup values based on a key
- In this case, can lookup based on probability of an event
- TABLE()
- Allows replication of a model with stochastic components to see the variability of the results after a large number of
- STDEV(result list)

Random Number Generation

An algorithm for producing numbers between 0 and 1.

0 £ x < 1

Excel: “=RAND()”

These numbers are generated uniformly – an equal probability of obtaining a number in any range over the interval. (rectangular distribution)

Alternative distributions are obtained by plugging the random number into other distribution functions, for example:

For example, to get a normal distribution:

Excel: =NORMINV(RAND(), 0,1)

Pr(x)

0

1

Converting RAND to various ranges

- RAND() generates numbers between 0 and 1.
- What if you want a different range? (still with Uniform distribution)
- Easy:
- MIN + RANGE * RAND()
- E.g.
- Probability ranges from .6 to .8
- .6 + .2*RAND()
- Payoff ranges from 1000 to 1500
- 1000 + 500*rand()
- Expected time is 8 minutes, plus or minus two minutes
- Min in this case is 6, max is 10

VLOOKUP and Custom /Discrete Probability Distributions

- We may have a discrete distribution which is relatively well-known
- We want to map from a uniform random to a custom distribution
- VLOOKUP allows us to lookup a random value in a list of cumulative probabilities and return the variable value.
- For example, say that demand level probabilities are known to be: Lookup Table:
- We could look VLOOKUP to randomly generate demand levels given this distribution:
- VLOOKUP(RAND(), LookupTable, 2)
- This tells Excel to generate random number between 0 and 1 (which we interpret as a probability), then lookup the first value in the lookup table equal to that value, and return the value in the second column.
- If no equal value is found, Excel returns the highest value less than the lookup value
- For example, a random value of .15 returns 100, .5 returns 200, .99 returns 300

Data Tables Command

- Used to replicate basic Excel model with random variation of some contributing variables
- Build a standard Excel model that captures the key attributes of the model (perhaps with static values)
- Allow for some element of randomness (RAND) in the model for the stochastic variables
- One-way data table:
- “Anchor cell” is the value or values you want to capture in the replication
- Top left cell, or top row of Data Table
- Select data range:
- Rows – R – number of replications
- Columns – C – number of captured statistics for each replication
- Data – Tables –
- Column Input Cell – any BLANK cell in spreadsheet
- Excel replicates your base model with R replications
- Create summary statistics of the captured key measures

Data Tables Continued

- Two-way Data tables
- If you want to test the benefits of various policy/decision variables in the face of uncertainty
- Anchor Cell – top left cell of data table
- Policy variable values – top row of data table
- Replication Count – Left hand column of data table
- Data – tables
- Row Input Cell – The cell in your BASE MODEL that you want to replace with the values in the top row of your data table
- Column Input Cell – any BLANK CELL in your spreadsheet
- Result – values of your anchor cell in each replication (row) for each policy variable value (column) you tested
- Take sample statistics on each column to evaluate the performance of each policy in the face of volatility.

Spreadsheet running slow?

- Data Tables are incredibly memory and CPU intensive.
- RAND() recalculates every time you change a cell
- Between the tow, you can spend a lot of time waiting.
- To fix this, choose Tools -> Options-> Calculations
- Change it from “Automatic” to “Automatic Except Tables”
- Then you hit F9 to recalculate tables, only when you want to.

Basic Statistics

- The range of outcomes depends on the min and the max observations, which are often outliers
- We may want to know what the interval is that profit will end up in, say, 95% of the time
- This depends on the standard deviation of the variable you are measuring
- The standard deviation of the mean of a random variable is:
- Std Dev (mean) = std dev / Sqrt (count of obs)
- The confidence interval is given by the z-score, where, for example,
- 95% of the sample means will lie within 1.96 standard deviations of the true mean
- We can estimate the 95% confidence interval as 1.96 standard deviations of the sample mean
- To compare the difference of two means with unknown variance, we use a t-test
- Tools – data analysis – t-test with unequal variances

Simulation of Inventory with Uncertain Demand

- Newsvendor Problem
- Given a mean demand of 100 papers per day, normally distributed with a standard deviation of 12
- The cost per paper is $.10; The Revenue per paper is $.30. The paper is worthless if it isn’t sold.
- What is the optimal level of inventory to hold?
- What is the 95% confidence interval of the mean profitability?

Operations Simulation

- Say you run a bakery and want to establish your customer service levels.
- You have a single employee who works the counter
- You know on average a customer arrives every 7 minutes with a standard deviation of 2 minutes
- You know that on average it takes 6 minutes to serve a customer with a standard deviation of 2 minutes
- Assume a normal distribution of inter-arrival times and processing times (there are better assumptions than this)
- Will your customers ever wait?
- If so, how long? What is the worst case?
- What is the overall utilization of your employee (time spent serving customers over total time)

Demand Simulation and Optimal Pricing

- We make optimal pricing decisions based on estimated regression estimates and cost curves
- Estimate the optimal price given demand and total cost data
- Regression: Quantity = Intercept + a*Price
- Regression: TC = Intercept + b*Quantity
- What is the optimal price and quantity
- Non-linear (quadratic) optimization: = P*(Intercept + a*Price)
- Objective: Profit = TR – TC;
- Decision Variable is Price; TR
- Unconstrained
- Test the robustness of this optimal solution, given
- Random variation in the estimated demand parameters
- Assume Normal distribution of price coefficient estimate (a)
- Note: Intercept = average(Quantity) – a*average(Price)
- Standard Error of Estimate indicates
- What is the 95% confidence interval on this solution?

A good model is one that...

- includes ONLY relevant information.
- includes ALL relevant information.
- is a valid representation of the system.
- provides meaningful & intelligible results.
- is quick and inexpensive to build.
- can be easily modified and expanded.
- is credible and convincing.

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