21 Decision Uncertainty

.title[Decision-Making Under Uncertainty] .subtitle[BEE 4750/5750] .subtitle[Environmental Systems Analysis, Fall 2022] <hr> .author[Vivek Srikrishnan] .date[November 14, 2022]

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# Outline

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1. Project Schedule
2. Questions?
3. Approaches to Decision-Making Under Uncertainty
4. Scenario Trees
5. Including Scenarios Into Optimization

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# Poll

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URL: [https://pollev.com/vsrikrish](https://pollev.com/vsrikrish)

Text: **VSRIKRISH** to 22333, then message]

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***Any questions?***

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# Last Class

<hr>

* Overview of Sensitivity Analysis
  * Application of Morris and Sobol Methods

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# Decision-Making Under Uncertainty

<hr>

Many optimization frameworks (LP, etc) assume deterministic problem formulations.

But most systems problems aren't *actually* deterministic.

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# Approaches to Decision-Making Under Uncertainty

<hr>

1. Solve deterministic problem using best estimates and/or expected values.
2. Stress-test with Monte Carlo (robustness, etc) or include safety margins.

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1. Solve stochastic problem for best expected performance (or similar summary statistic).

]

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# Representing Uncertainty in Decision Problems

<hr>

*Previously with the Lake Problem*: Uncertainty in non-point source inflows represented through a distribution.

In this case, we can do Monte Carlo analysis to estimate some summary statistic:

* Expected performance
  * Reliability of strategy

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# Representing Uncertainty in Decision Problems

<hr>

But we can also represent uncertainty through scenarios and sequential events, which sometimes lets us rewrite decision problems to directly incorporate different outcomes.

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# Revisiting the Farmer Problem

<hr>

A farmer can grow wheat, corn, and sugar beets on 500 ha of land. How much land should they allocate to each crop?

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# Revisiting the Farmer Problem

<hr>

Key information:

* Planting costs are $\$150$, $\$230$, and $\$260$ per acre for wheat, corn, and sugar beets, respectively.
  * The farmer requires $200$T of wheat and $240$T kg of corn for feed, and excess can be sold for $\$170$ and $\$150$ per T, respectively.
  * Purchasing crops costs $40\%$ more than the selling prices.
  * Sugar beets are sold at $\$36$/T up to $6000$T, at which point the price drops to $\$10$ per T.

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# Farm Yields

<hr>

From prior experience, the farmer knows that in an average year, yields are:

* 2.5 T/acre for wheat;
  * 3 T/acre per corn;
  * 20T/acre for beets.

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# Problem Formulation

<hr>

Decision variables:

| Variable | Definition                |
|:--------:|:------------------------- |
|  $x_i$   | acres of crop $i$ planted |
|  $y_i$   | T of crop $i$ purchased   |
|  $z_i$   | T of crop $i$ sold        |

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# Deterministic Problem Formulation

<hr>

$$
\begin{alignedat}{3}
&\min\_{x, y, z} & & 150x\_1 + 230x\_2 + 260x\_3 +  238y\_1 + 210y\_2 - 170 z\_1 \\\\[0.5ex]
& &&\qquad -150z\_2 - 36z\_3 - 10z\_4 \\\\
& \text{subject to:} & & \qquad\\\\
& && x\_1 + x\_2 + x\_3 \leq 500 \\\\
& && 2.5 x\_1 + y\_1 - z\_1 \geq 200 \\\\
& && 3 x\_2 + y\_2 - z\_2 \geq 240 \\\\
& && z\_3 + z\_4 \leq 20 x\_3 \\\\
& && z\_3 \leq 6000 \\\\
& && x\_i, y\_i, z\_i \geq 0
\end{alignedat}
$$

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# Deterministic Solution

<hr>

| Variable      | Wheat | Corn | Beets |
|:------------- |:-----:|:----:|:-----:|
| Area (acres)  |  120  |  80  |  300  |
| Yield (T)     |  300  | 240  | 6000  |
| Sales (T)     |  100  |  –   | 6000  |
| Purchased (T) |   –   |  –   |   –   |

This solution yields a profit of $\$118,600$.

What is this solution doing? Does it make sense?

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# Now Let's Add Uncertainty

<hr>

But yields tend to vary from year to year. From prior experience, the historical variability is:

* in a good year, yields can be 20% above average;
  * in a bad year, yields can be 20% below average.

To simplify things, let's assume these yields vary consistently across crops and that each scenario (average, good, bad) has an equal probability of occurrence.

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# Stochastic Formulation

<hr>

What should our decision variables be?

]

| Variable | Definition                              |
|:--------:|:--------------------------------------- |
|  $x_i$   | acres of crop $i$ planted               |
| $y_{ij}$ | T of crop $i$ purchased in scenario $j$ |
| $z_{ij}$ | T of crop $i$ sold in scenario $j$      |

]

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# Stochastic Objective

<hr>

How can we formulate an objective?

**First choice: what statistic are we trying to optimize?**

Let's choose the expected value $\mathbb{E}\left[\text{Profit}\right]$.

Other possible choices: quantiles (*robust optimization*) to hedge against worst-case outcomes, variance (to minimize year-on-year fluctuations).

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# Scenario Tree for Objective

<hr>

Since we have a discrete (and small) set of outcomes, we can use a **scenario tree** to write out the different outcomes.

This is overkill for this problem, but can be useful when there are nested or sequential outcomes (we'll see examples later). ]

]

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# Stochastic Objective

<hr>

Our new objective becomes:

$$
\begin{alignedat}{2}
&\min\_{x, y, z} & & \quad 150x\_1 + 230x\_2 + 260x\_3 \\\\[0.5ex]
& &&\qquad -\frac{1}{3} \left(170z\_{11} + 150z\_{21} + 36z\_{31} + 10z\_{41} - 238y\_{11} - 210 y\_{21}\right) \\\\[0.5ex]
& &&\qquad -\frac{1}{3} \left(170z\_{12} + 150z\_{22} + 36z\_{32} + 10z\_{42} - 238y\_{12} - 210 y\_{22}\right) \\\\[0.5ex]
& &&\qquad -\frac{1}{3} \left(170z\_{13} + 150z\_{23} + 36z\_{33} + 10z\_{43} - 238y\_{13} - 210 y\_{23}\right)
\end{alignedat}
$$

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# Stochastic Constraints

<hr>

$$
\begin{alignedat}{2}
&x\_1 + x\_2 + x\_3 \leq 500 \\\\
&\color{blue}3x\_1 + y\_{11} - z\_{11} \geq 200, && \qquad  \color{blue}3.6x\_2 + y\_{21} - z\_{21} \geq 240 \\\\
&\color{blue}z\_{31} + z\_{41} \leq 24x\_3, && \qquad \color{blue}z\_{31} \leq 6000 \\\\
&\color{purple}2.5x\_1 + y\_{12} - z\_{12} \geq 200, && \qquad \color{purple}3x\_2 + y\_{22} - z\_{22} \geq 240 \\\\
&\color{purple}z\_{32} + z\_{42} \leq 20x\_3, && \qquad \color{purple}z\_{32} \leq 6000 \\\\
&\color{red}2x\_1 + y\_{13} - z\_{13} \geq 200, && \qquad \color{red}2.4x\_2 + y\_{23} - z\_{23} \geq 240 \\\\
&\color{red}z\_{33} + z\_{43} \leq 16x\_3, && \qquad \color{red}z\_{33} \leq 6000 \\\\
&x\_i, y\_i, z\_i \geq 0
\end{alignedat}
$$

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# Stochastic Solution

<hr>

| Variable                   | Wheat | Corn | Beets |
|:-------------------------- |:-----:|:----:|:-----:|
| Deterministic Area (acres) |  120  |  80  |  300  |
| Stochastic Area (acres)    |  170  |  80  |  250  |

Deterministic Solution: $\text{``Expected'' Profit (no uncertainty)} = \$118,600$. Stochastic Solution: $\mathbb{E}[\text{Profit}] = \$108,390$.

**What should we do?**

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# Expected Performance of Deterministic Solution

<hr>

Note that the deterministic solution only gives us the expected payoff in the average year: we'd get more or less yield in other years, which would affect profits.

| Year    | Deterministic Profit | Stochastic Profit |
|:------- |:--------------------:|:-----------------:|
| Good    |       $148,000       |     $167,000      |
| Average |       $118,600       |     $109,350      |
| Bad     |       $56,800        |      $48,820      |

**Expected Profit from Deterministic *under uncertainty***: $107,240

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# Value of the Stochastic Solution

<hr>

We can now compare the difference in expected profits *under uncertainty*, which is the fair comparison. This is called the **value of the stochastic solution (VSS)**, or sometimes the **expected value of including uncertainty (EVIU)**.

$$
\begin{aligned}
VSS &= \mathbb{E}[\text{Stochastic Profit}] - \mathbb{E}[\text{Deterministic Profit}] \\\\
&= \$108,390 - \$107,240 \\\\
&= \$1,150.
\end{aligned}
$$

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# Sensitivity of the VSS

<hr>

**Note**: The obtained stochastic solution (and its value) can be highly sensitive to scenario probabilities!

For example: suppose the probability of a good harvest is 1/8, an average harvest is 1/8, and a bad harvest is 3/4.

| Crop  | Area (acre) |
|:----- | -----------:|
| Wheat |         100 |
| Corn  |         100 |
| Beets |         300 |

]

| Year    | Deterministic Profit | Stochastic Profit |
|:------- | --------------------:| -----------------:|
| Good    |             $148,000 |      **$147,000** |
| Average |             $118,600 |      **$117,500** |
| Bad     |              $55,120 |       **$56,800** |

]

The VSS is then **$998**.

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# Sensitivity of the VSS

<hr>

Now suppose the probability of a good harvest is 3/4, an average harvest is 1/8, and a bad harvest is 1/8.

| Crop  | Area (acre) |
|:----- | -----------:|
| Wheat |         183 |
| Corn  |          67 |
| Beets |         250 |

]

| Year    | Deterministic Profit | Stochastic Profit |
|:------- | --------------------:| -----------------:|
| Good    |             $148,000 |      **$167,667** |
| Average |             $118,600 |      **$107,683** |
| Bad     |              $55,120 |       **$47,670** |

]

The VSS is then **$12,458**.

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# Value of the Stochastic Solution

<hr>

**To summarize**:

* A solution obtained assuming deterministic outcomes might "on paper" yield better anticipated outcomes.
  * Solutions to stochastic problems can produce a better expected outcome than the deterministic solution, but will perform better/worse in any given scenario.
  * Stochastic solutions can be highly sensitive to scenario probabilities; be careful about deep uncertainty.

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# Expected Value of Perfect Information

<hr>

Another relevant quantity, the **expected value of perfect information (EVPI)**, concerns the value associated with better forecasts.

If the farmer had perfect foresight and could allocate acreage $x$ accordingly and sell/purchase $z$ and $y$ optimally (versus taking the 1/3-probability stochastic solution),

$$
\mathbb{E}[\text{Profits} | \text{perfect information}] = \sum_j p_j \times \text{Profits}_j = \$115,405.
$$

$$
\text{EVPI} = \$115,405 - \$108,390 = \$7,015
$$

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# Other Approaches to Incorporating Uncertainty

<hr>

* Chance-Constrained Optimization
  * Robust Optimization
  * Stochastic Dynamic Programming
  * Simulation-Optimization with Monte Carlo estimates of objective

**Note**: All of these can get very computationally expensive *very* quickly.

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# Simplified Reservoir Operations Example

<hr>

Consider two conflicting objectives when managing a reservoir:

* Want to avoid floods
  * Want to maintain appropriate water supply.

With a high water level, water demand is likely to be met, but highly damaging flooding is more probable.

With a low water level, flooding is less likely, but also less likely to meet demand.

**To Be Continued!**

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# Key Takeaways

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* Ignoring uncertainty may *accidentally* lead to better solutions, but we would like to incorporate uncertainty into decision-making when possible.
  * Value of the stochastic solution quantifies expected improvement in performance.
  * Need to balance computational expense with representation of uncertainty.

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# Next Class

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* More examples of decision-making under uncertainty