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In the chapters on linear, integer, and nonlinear
optimization, we used deterministic models.
In most situations, some of the data will be
uncertain, which implies inherent risk.
Stochastic models incorporate uncertainty.
If an optimization model has uncertain variables,
we might first solve it deterministically and then
use Monte Carlo simulation to analyze the results.
The Sklenka Ski model (Chapter 13), seeks to maximize
profit subject to constraints on:
- Fabrication labor hours
- Finishing labor hours
- Market mixture
Suppose the labor hours required for finishing is
stochastic; then overtime will be needed if more than 21
hours of finishing time are required.
 Finishing time will be modeled by triangular distributions.
How often will overtime be needed if the optimal solution
of 5.25 Jordanelle and 10.5 Deercrest skis are
scheduled each day?
Spreadsheet model
Specify the finishing time input
distributions in cells B7 and C7.
Specify the available finishing hours
as an uncertain output cell D16.
Analytic Solver Platform Simulation Results
The likelihood of
needing overtime
is about 85%
A chance constraint is one that specifies the fraction of
trials in a simulation that must satisfy a constraint.
Suppose that the company wants to determine a daily
schedule so that the probability of overtime—that is,
requiring more than 21 hours of finishing time—is less
than 0.1, or 10% of the time.
◦ In this case, we would want to specify that the percentage of trials
requiring less than 21 hours of finishing time is at least 90%.
Chance constraints are defined by a percentile, or value
at risk (VaR), measure.
A VaR constraints with chance p% requires that the
constraint be satisfied p% of the time.
◦ This does not consider the magnitude of the violation when the
constraint is not satisfied.
Conditional at risk (CVaR) constraints place bounds
on the average magnitude of all violations of the
constraint that may occur (1−p)% of the time.
◦ CVaR is more conservative than VaR.
Sklenka Skis
wants to
determine a
schedule that
has no more
than a 10%
probability of
overtime being
required. That is,
they want a 90%
probability of
needing 21 or
fewer hours of
finishing labor.
Solution with chance constraint
Simulation results with chance constraint
Solver typically finds a conservative solution to problems with
chance constraints. However, Analytic Solver Platform can
automatically improve the solution by adjusting the size of the
uncertainty set for the chance constraint auto-adjust process.
The standard EOQ model assumes constant
(deterministic) demand. In most practical
situations, demand is stochastic.
If D is uncertain, then the demand during the lead
time will also be uncertain. This impacts how the
reorder point should be chosen.
We can use Monte Carlo simulation to analyze the
optimal solution.
EOQ Example A.5:
Annual demand = 15,000 units.
Ordering costs = $200 per order.
Purchase cost = $22 per item.
Carrying charge rate = 20%.
Assume demand is normally distributed with a
mean of 15,000 units and a standard deviation of
2,000 units.
Spreadsheet model
Cell B5 is defined to be normally
distributed using the function
=PsiNormal(15000, 2000).
In cell B22, we calculate the leadtime demand by multiplying the
annual demand rate (B13) by the
lead time in B21 and define it as an
output cell.
Distribution of lead-time demand
A service level is a constraint that represents the
probability that demand can be satisfied.
◦ For example, we might want to ensure that demand can
be satisfied 95% of the time.
We can identify the reorder point for a particular
service level from the frequency chart.
In the distribution of lead time demand, first set the Lower Cutoff
value in the Chart Statistics pane to zero and then set the Likelihood
value to 0.95. This will calculate the Upper Cutoff value so that the
cumulative probability is 0.95.
A 95% service level
requires a reorder
point of about 351
units. The additional
351 - 288 = 63
units above the
mean lead-time
demand is called
safety stock.
Price-demand elasticities are only estimates and
most likely are quite uncertain.
Assume that the true values might vary from the
estimates by plus or minus 25%. Model the
elasticities by uniform distributions.
Using the optimal prices identified by Solver,
use Monte Carlo simulation to see what
happens to the prediction of the number of
rooms sold under this assumption.
Spreadsheet model
Output cell
Simulation results for 450-room capacity
By changing the Upper Cutoff value in the task pane, we
could identify the likelihood of exceeding that value. the
likelihood of exceeding 457 rooms is close to 10%.
Shift the capacity constraint down by 7 rooms to 443 and
find the optimal prices associated with this constraint, we
would expect demand to exceed 450 at most 10% of the
Confirmation simulation run
Analytic Solver Platform provides a capability –
called multiple parameterized simulations - of
automatically running simulations for a range of
values for decision variables.
 In the Newsvendor Model, for example, we can vary
purchase quantities of the candy boxes to determine the
optimal number to purchase.
 In the Hotel Overbooking Model, we can find the best
number of reservations to accept.
Newsvendor Model with Historical Data
First, set the demand in cell B11 =PsiDisUniform(D2:D21). Then
select cell B12 and set a lower limit of 40 and upper limit of 51 in
the Function Arguments dialog (see text for further implementation
details). Analytic Solver Platform will run 12 simulations for each
purchase quantity.
Now we want to find the optimal purchase quantity
by varying purchase quantity between 40 and 51.
Select cell B12.
Risk Solver
Values or Lower: 40
Upper: 51
All Options
Simulations to Run: 12
Hotel Overbooking Monte Carlo Simulation Model with
Custom Demand
See text for implementation details. Solver identifies 313
reservations as the best solution, just as we found using
the multiple parameterized simulation approach.
An investor has $100,000 to invest in four assets. The expected
annual returns and minimum and maximum amounts with which the
investor will be comfortable allocating to each investment are:
Arbitrate pricing theory provides estimates of the sensitivity of
investments to risk factors such as inflation, industrial production,
interest rates, etc.
Risk factors
Determine how much to invest in each asset to maximize the total
expected annual return, remain within the minimum and maximum
limits for each investment, and meet the limitation on the weighted
risk per dollar invested (assumed to be 1.0).
Define Xi as the amount invested in asset i
Maximize 0.05X1 + 0.07X2 + 0.11X3 + 0.04X4
X1 + X2 + X3 + X4 ≤ 100,000
− 0.5X1 + 1.8X2 + 2.1X3 − 0.3X4 ≤ 1.0(X1 + X2 + X3 + X4)
2,500 ≤ X1 ≤ 5,000
X2 ≥ 30,000
X3 ≥ 15,000
X4 ≥ 0
Assume that annual returns are uncertain for all
but the savings account.
Life insurance returns are uniformly distributed.
 Cell B6: =PsiUniform(4%, 6%)
Bond mutual fund returns are normally distributed.
 Cell B7: = PsiNormal(7%, 1%)
Stock fund returns are lognormally distributed.
 Cell B8: = PsiLogNormal(11%, 4%)
Also, define Cell D20 (total expected return) as an
uncertain output cell by adding +PsiOutput()
Simulation of the expected return
Project-selection and capital-budgeting projects
typically have many uncertainties because they
involve future events.
 Returns and resource requirements are often uncertain
Implementing a project is not guarantee of
successful completion.
Analytic Solver Platform allows for the
incorporation of uncertainties in project selection
Example 15.5 (Hahn Engineering Project Selection)
Expected returns are uncertain and can be modeled
using lognormal distributions.
Also, assume that some projects are riskier than others
and have different probabilities of being completed
To model whether a project is successful, use a
binomial probability distribution with n = 1.
Use IF statements to apply the returns and
success probabilities only to “selected” projects.
Specify total return as a changing output cell.
See text for implementation details.
Model and solution
Simulation results