Stochastic Programming

1
AED Economics 702 Computational
Economics Decision Making With Uncertainty and
Stochastic Programming Linus Schrage, Chapter 12.
2
Chance Constrained Programming

• The Problem In Words
• You must determine how much of four available
  grains to include in blended feed (in
  circumstances under which the content of one of
  the nutrients varies at random) to meet nutrient
  requirements with some degree of certainty.
• Background
• You've had several samples of the four grains
  tested and you find that their content of
  Nutrient D varies randomly and is normally
  distributed. You want to find a new blend that
  ensures the minimum requirements for Nutrient D
  are met at least 95 of the time.

3
Standard Normal Distribution

• A refresher on the Z random variable

4
Chance or Risk Modeled as Mean/Variance

• You've calculated the mean and variance for each
  grain (the content means are the same as in the
  original HOGFEED model). The equation that
  calculates Nutrient D now looks like this
• (Mean Nutrient D in the blend) - Z(standard
  deviation Nutrient D) gt 21.
• For 95 confidence in the content of Nutrient D,
  set Z 1.645 (Refer to any elementary text on
  statistics for a discussion of Z values).

5
Whats Best! and the Hog Feed Model

• Objective of Optimization
• The objective is to determine how much of each
  grain you should buy at today's prices to meet
  their nutritional requirements at lowest cost.
• Specify Constraints
• The limitation to this problem is that the final
  mix must contain the minimum required levels of
  nutrients for Swine Roses' hogs. This
  limitation is enforced by creating a constraint
  cell for each nutrient (H7H9, H11). Each
  constraint will return the "Not gt" indicator
  until the Minimum Required (I7I9, I11) for that
  nutrient is met in the Nutrients Supplied column
  (G7G9, G11).

6
A Note on Dual Values in NonLinear Models

• DUAL VALUE and NonLinear Models
• In nonlinear models, the ranges over which dual
  values are valid may be very small. Before basing
  a pricing or purchasing decision on a dual price
  or reduced cost in a nonlinear model, you should
  test the returned value by making the specified
  change to the variable or constraint, using
  AdjustableRemove Adjustable on the variable, and
  re-solving.

7
Swine and Roses Hog Feed Model

• Cell G11 SUMPRODUCT(C11F11,C18F18)
  -1.645(C12C182D12D182E12E182F12F182)0
  .5

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HogChance Whats Best Model
9
Peanut Production Quota
10
Peanut Production Quota
11
Peanut Production Quota
12
Peanut Production and Quota
13
Peanut Production and Quota
14
Peanut Production and Quota
15
Peanut Production and Quota
16
Peanut Production and Quota
17
Peanut Production and Quota
18
Mathematical Programming and Modeling Uncertainty

• Key aspect of the Uncertainty model is the
  multi-period nature of the decision process.
• Decisions in period t have alternative outcomes
  with then influence decisions in period ti
• Type of Uncertainty
• Weather related
• Inventory models
• Production models
• Financial
• Market price movements
• Loan repayment or default

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Types of Uncertainty

• Political Events
• Changes in government regime
• Outbreaks of war / hostilities
• Technology
• Availability of new technology when needed
• Equipment failure models
• Market Events
• Shifts in consumer preferences
• Population shifts
• Competition
• Game theory models
• Strategic behavior by competitors

20
Risk Modeling and Programming

• Why model risk in a programming problem?
• Why not solve the model under all combinations of
  the risky parameters and used these solutions?
• Dimensionality consider that five values for 3
  parameters requires 35 243 possible
  combinations of parameter specifications!
• Certainty each of the 243 solutions is
  optimal given that you are certain of the values
  of the parameters.
• Risk Modeling Approaches attempt to provide a
  solution that is satisfactory across a
  distribution of parameter values.

21
Risk Modeling and the Decision Horizon

• Two fundamental situations arise in risk
  modeling
• One all decisions must be made NOW with the
  uncertain outcomes resolved later, after all
  random draws from the distribution are known.
• This type of risk modeling is represented by
  Stochastic Programming without Recourse.
• Two some decisions are made now, then later
  some uncertainties are resolved, and then another
  set of decisions must be made.
• This type of risk modeling is represented by
  Stochastic Programming with Recourse.

22
Uncertainty and Multi-Period Analysis

• Two Period Model
• Make a first-period decision
• State of Nature is determined
• Make a second period decision with information
  from the outcome of the first period

23
Two Period Model Graphically
CORN(ct)
EP
289
283
CORN(nt)
-55
0.88
-101
24
Two Period Model Graphically
CORN(nt)
EP
289
BEANS(nt)
112
90
-55
0.45
25
Two Period Model Outcomes

• Expected Profit Positions
• P is the probability of Adequate Moisture
• Corn(ct) -101 390 p
• (289)p -101(1-p) 289p 101 101p
• Corn(nt) -55 338 p
• (283)p - 55(1-p) 283p - 55 55p
• Beans(ct) 41 77 p
• (118)p 41(1-p) 118p 41 - 41p
• Beans(nt) 90 22 p
• (112)p 90(1-p) 112p 90 90 p

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Two Period Model Outcomes

• Objective Maximize Expected Profit
• Take the probability of an AM season at 3/8
• Max 3/8 (289Cct 283Cnt 118Bct 112Bnt)
• 5/8 (-101Cct 55Cnt 41Bct 90Bnt)
• St Cct Cnt Bct Bnt lt 1
• What is the solution? Why?
• 31.25 C 51.25 W 51.875 B
• St C S B lt 1
• Solution is to plant all beans!

27
Whats Best! MODEL AND SOLUTION
28
LINGO MODEL AND SOLUTION
29
LINGO MODEL AND SOLUTION
30
Two Period Snow Removal Problem
31
Two Period Snow Removal Problem

• Winter is classified as Warm or Cold (states of
  nature)
• Warm with probability 0.40
• Cold with probability 0.6
• Decisions made before Winter are Period 1
• Decisions made during or after Winter are Period
  2
• Truck day is the amount consumed by one truck in
  one day
• Period 2 price of salt is a random variable
• Higher in a cold winter

32
Two Period Snow Removal Problem

• Operating cost of a truck/day depends on state of
  nature (warm 110 / cold 120)
• Truck fleet capacity is 5000 truck-days
• Plowing only requires 3,500 tds in warm winter
  and 5,100 in cold winter
• Salting is efficient Warm winter 1 1.2 Cold
  1 1.1
• In a cold winter some salting will be necessary
  due to limited truck capacity

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Define the Variables for this problem
34
Define the Variables for this problem
35
The LINGO Model Warm
If we know that the winter will be warm (state of
nature warm)
36
The LINGO Model Cold
If we know that the cold will be warm (state of
nature cold)
37
The LINGO Model Sets Version Cold
38
The LINGO Model Solution Cold
In a cold winter the most efficient is to plow.
But there is a binding constraint on truck
capacity so just enough salting is used to make
the truck capacity sufficient. Can you identify
this in the output ??
39
The Conditional and Unconditional Solution

• The Warm model and Cold model are conditional
  models, i.e, conditional on knowing ahead whether
  the winter will be warm or cold. Each has a
  different solution for first period purchasing.
• The unconditional model will combine the Warm and
  Cold models into one model and use probabilities
  to influence the optimal solution.
• The same first stage variables appear in both
  sets of constraints forcing the first stage
  decision regardless of the outcome of the winter.
• What needs to be completed is the specification
  of the appropriate objective function.

40
The combined Objective function

• First period cost coefficients are correct in the
  Warm and Cold models.
• Second period costs, KW and KC must be treated as
  random variables.
• KW applies with probability 0.40
• KC applies with probability 0.60
• The objective function becomes
• Min z (70 BF1) (20 BS1) 0.4 KW
  0.6 KC

41
The LINGO Combined Model
Z Min 70 BF1 20 BS1 0.4 KW 0.6 KC
42
Summary Two Period Planning with Uncertainty

• Build a complete model for each state of nature
• Combine these models into one unconditional model
• First stage (period) variables must be common to
  all submodels
• Second stage (period) variables appear only in
  the appropriate submodel
• Second stage cost for each submodel appears in
  the overall objective function with weights that
  reflect the probabilities that nature will select
  the state corresponding to that submodel.
• Use the Range Report to examine the sensitivity
  of the model solution to the probabilities
  selected for each state of nature.

43
The Combined Model Solution
What does the model tell us about our decision
over the two periods ?? 1 Purchase sufficient
salt and fuel in period 1 to follow a pure
salting policy if the state of nature is WARM. 2
If the state of nature is COLD, extra fuel is
purchased in the second period to be used for
plowing at the margin. 3 With this solution
there is NO excess salt or fuel at the end of
period 2. What is the impact of a projection of a
higher probability of a WARM winter than that
reflected in this problem? How could we evaluate
this event?
44
Range Analysis for KW and KC
45
RHS Ranges What can we learn ?
46
Expected Value of Perfect Information

• We have learned from the WARM and COLD models
  what the least cost solution is for our uncertain
  outlook. Now what is the value of having perfect
  foresight? Perfect information?
• Warm Winter Cost 583,333
• Cold Winter Cost 970,000
• Perfect Forecasts imply over time a weighted
  average of costs
• 0.4 x 583,333 0.6 x 970,000 815,333
• The Unconditional model solution is 813,333

47
The Expected Value of Perfect Information

• 815,333.3 - 813,333.3 2,000.00
• This is the maximum we should pay for a
  prediction or forecast on the upcoming weather
  for the winter.

48
Expected Value of Modeling Uncertainty

• Certainty Equivalence Theorem
• If the randomness or unpredictability exists
  solely in the objective function coefficients,
  then it is correct to solve the LP after simply
  using the expected values for the random
  coefficients in the objective function.
• If the randomness exists in the RHS or constraint
  coefficient(s) then it is NOT correct to replace
  these with the expected value.
• (see the rules in Schrage, page 329) and the
  planting example.

49
Certainty Equivalence Rules

• The model variables

The random variable Y can be replaced by its
expected value E(Y) if Y appears only in the
objective function, and each term containing Y
is not a function of X, or is linear in Y and
contains no random variables dependent upon Y
50
Risk Aversion in Modeling Uncertainty

• In the snow removal problem the cost of period 1
  and period 2 977,591 - IF the winter is
  cold.
• IF it is known that the winter will be cold then
  the cost is 970,000.
• Risk aversion can be incorporated into the model
  by adding the constraint
• 70 BF1 20 BS1 KC lt 975,000.
• Why does this reflect a level of risk aversion?
• What is the solution to this model if this
  constraint is incorporated into the combined
  model?
• Expected cost increases by 173.

51
Downside Risk Modeling

• A look at risk or the quantification of risk by
  considering returns that are lower than some
  threshold.
• Downside risk is the expected amount by which a
  return falls short of a specified target level.

52
Define a Downside Risk Model
53
Reconsider the Corn / Soybean / Sorghum Model

• The farmer eliminates soybeans from the decision
  variables
• Increases the probability of a wet season to 0.7
• The revised model

54
The LINGO Solution

• Solution is to plant 100 Corn with an expected
  profit of 67
• If the season is dry, profits RD will be negative
• Compute the expected downside risk for this
  simple model
• Select a target threshold
• A conservative target is a value of 40 for a zero
  downside risk on sorghum

55
Downside Risk Constraints
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The LINGO Downside Risk Model
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The LINGO Downside Risk Solution
What is the expected downside risk for this model?
58
Increase the level of Risk Aversion
Add the constraint ER lt 10
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What happens with ER lt 10??

• Optimal solution is to put 1/3 of land in Sorghum
• Profit declines from 67 to 65
• If we draw a dry season then profit will be 6.67
  and not 10 as before
• As we increase the level of risk aversion, e.g.,
  we change ER from lt 10 to 0 the amount of land
  devoted to sorghum increases and the amount
  devoted to corn declines.

60
Chance Constrained Programming

• The dynamic multi-period approach can grow quite
  large is the number of possible states of nature
  is large
• If n periods and s equals states then the
  problem is proportional to sn.
• Chance Constrained Programming is an approach to
  solve this dimensionality problem.
• Stochastic programs require that every constraint
  be satisfied by some combination of first and
  second period decisions
• Chance Constrained programs allow each
  constraint to be violated with a certain
  specified probability.

61
Chance-constrained Snow Removal Problem

• Eliminate the second stage decision variables
• Must specify a probability allowance for each
  constraint
• The snow removal problem redefined
• P 0.75 probability of providing required snow
  removal capacity for the severity of the winter
• Must provide 5,100 TDs of snow removal capacity
• One truck day of operation is 116
• One truck day of salting equals 1.14 TDs of
  plowing
• The Chance-constrained LP model

62
Chance-constrained Snow Removal Model
63
Chance Constrained Programming

• Marketing of Cotton Fiber in the Presence of
  Yield and Price Risk. Paper presented at the
  Southern Agricultural Ag Economics Association
  1999.
• Expected Utility Model
• Chance constrained linear programming
• Analyze four marketing strategies and seven crop
  insurance alternatives.
• General Conclusion Existing marketing tools and
  crop insurance products can replace government
  support programs for cotton.