# Robust optimization in timberland and wood industry analyses

**This post describes an overview and applications of robust optimization technique in the context of timberland and wood industry analyses. **

# Background

Forests are complex systems, what makes that forestry-related decisions face risk and uncertainty. The decisions in forestery often concern large areas, long time horizons and multiple stakeholders, what makes also forest management planning, predictions and assessments even more complex.

Forest sector modeling combines all above issues together into one framework, and adds even more complexities, uncertainties and risks. In forest sector models (FSM)*, *we have not only uncetain information about forestry, but also we have to deal with forest industries, and market interactions between forests and forest products. One can say that risk and uncertainty in FSM (but also in portfolio optimization or harvest scheduling models), are everywhere, especially in:

- model structure and its parameters such as demand elasticities, input–output coefficients and forest growth parameters,
- the quality of data describing the past and current state of the world,
- exogenous assumptions such as population and GDP growth rates or demand changes.

**In any decision, based on models, analyses or any other tools, the target is one: t o improve decision making. **

**>>READ ALSO: How to make better decisions in the forest sector?**

# Risk vs. uncertainty

Before we discuss robust optimization, first, we should clarify the difference between risk and uncertainty, as discussions about these two concepts have been ongoing for decades and there are many definitions. Shortly speaking, we talk about risk when the probability (objective or subjective) of factors is known. This implies that using the risk concept, one accepts that lack of objective probability can be replaced by subjective probability.

Uncertainty on the other hand, can be defined as lack of information, which may or may not be obtainable. Main sources of uncertainty are ignorance or events that have not yet occured. Few simple examples of uncertainty:

“The phonograph is of no commercial value” Thomas Edison, 1880

“There is no reason for any individual to have a computer in their home.” – Ken Olsen, President of DEC, 1977

Internal sales forecasts for PCs for the 1980s: 295,000. Actual sales for PCs in the 1980s: over 25,000,000 – IBM, 1979

“Why are you dodging like this? (to one of the soldiers in the Civil War) They couldn’t hit an elephant at that dist…” – Last words of General Sedgewick, 1864

It would make no meaning in numerical models to include parameters which we have no information about and cannot quantify. Risk as defined above, on the other hand, covers all cases where a probability distribution of the variable in question can be assumed. Let’s use the term risk then.

## Projections vs. Reality in softwood pulpwood prices in SE U.S.

In one of my studies, where I tried to predict softwood pulpwood prices in the Southeastern United States based on expected U.S. renewable policy and pellet export to EU, I have learned more about risk in such projection, when I compared them with actual situation that has happened on the market.

We can observe that historical prices have behaved completely different from my projections. Based on my present experience and conclusions made from previous study, we could improve our modelling framework, assumptions and consequently decision-making process. Such excercises help us definitely to understand the present and improve our methods to forecast future in more reliable way. But still, we should always be aware that predicting future based on the past and present, involves many risk factors. Nevertheless, there is no alternative to overcame all types of risk in forecasts, and we have to live with that, and look for other methods that can slightly improve such projections. Robust optimization, discussed here, is one of such approaches.

# Approaches to consider risk in forestry decision analysis

In order to reduce risk and improve decision making, **robust optimization** comes may appear helpful in certain cases.

All forest models may be divided on two large categories:

- deterministic (risk not explicity consiered), or
- stochastic

To analyze risk with deterministic studies, we can easily apply sensitivity analysis, scenario analysis or even Monte Carlo simulation. Deterministic models are more common that stochastic ones, as with stochastic models (depending on their complexities) the system generally is not solved analytically and it is extremaly difficult to interprete results. Robust optimization, in my opinion, helps us to find a bridge between deterministic and stochastic models, that can help us to limit risk in decision making and in the same time be solved analytically with output easy to understand.

## Simple idea behind Robust Optimization

Let’s see first simple linear programming problem, where we want to maximize some function given a budget constraint. We can imagine a wood broker, who want to maximize his objective function $3X + $2Y, where $3 is a price for spruce logs, and X is the number of spruce logs cubic meters (m3), and $2 is a price of pine logs and correspondigly Y is a volume of pine logs. The question is how many cubic meters of pine and spruce logs broker should buy to maximize his revenues. The answer normally would be as much as he can, but the problem is (like in real life) that the broker has to buy somewhere these logs (in order to trade) and he has a budget constraint given by equation $6X + $1Y≤ 6$, where the cost of spruce log is 6 dollars, pine 1 dollar and he has a budget of 6 dollars. Graphically, the problem look like below:

A green area show our budget possibility set, while the dashed green line is a revenue function that we want to maximize, given our feasible budget. After shifting the revenue curve, we are getting the solution, that broker needs to buy 6m3 of pine, and no logs of spruce to maximize his revenues, which are in this case $3*0 + $2*6 = 12 dollars. By buing 6m3 of pine he satisfy his budget constraint, e.g. $6*0 + $1*6≤ 6$.

The problem with this case is that the prices of logs that our broker buys may vary. Let’s assume for simplicity that prices for logs that he sells are constant (he may have a contract that specifies constant purchase price from the pulpmill). We have seen earliner in this post that pulpwood softwood logs in southeastern U.S. can vary over time, what creates some risk for the broaker. Therefore, in our set of equations we are not sure what will be the price the broker buys logs, i.e. they will not be always $6 and $1 per m3. For instance, such variations occur also in interest rates that are particularily important in forest business decisions.

**>>READ ALSO: Why interest rates differ in timberland investments?**

In order to perform, robust optimization we have to define something what is called “uncertainty set”. The “uncertainty set” has to contain all possible values for our spruce and pine log prices that satisfy all constraint equations in our equations set (here for simplicity we have only two constraint equations including this about non-negative values for amount of wood traded). Thanks to the uncertainty set we are able to draw many possible budget lines that fulfill constaint criteria. See graph below for better understanding.

In other words, the last clause “for all (c,d)” makes it a robust optimization problem because it implies that for a pair (X,Y) to be acceptable, the constraint cX + dY <= 6 must be satisfied for all values of (c,d) including the worst (c,d) pair that maximized the value of cX + dY for the given values of (X,Y). For this example, the “uncertainty set” is simplified to a finite set meaning that for each (c,d) within the set, there is a constraint cX + dY <= 6.

In order to find optimal solution under robust optimization and its “uncertainty set” we need to again repeat the procedure of shifting revenue line ($3X + $2Y) to cross the budget constraint frontier. Already now we can observe that our solution result in smaller amount of pine logs, and same as before – zero spruce logs. In other words, under all possible budget lines we selected the worst case scenario, that was driven by possible high prices of pine and spruce that may likely occur in the future. You can check it by substituting for example pine log purchase price of $1 by, let’s say vlaue of $20 – the result will be that the line will become flatter (keeping spruce logs price constant).

We selected the worst case scenario, which takes the highest possible purchase prices of logs that broker has to pay. And here is the point, the **robust optimization technique** is often called “**worst case analysis**“. You can call it “pessimistic approach” as well. Some people say that the difference between pessimists and optimists is that pessimists are always better informed. And robust optimization takes such approach 🙂

## How to obtain “uncertainty set”?

First quesiton you may ask is, how to define the uncertainty set, which is a crucial concept in the robust optimization method. Here, it may come the probability theory, which can guide us in the construction of uncertainty sets. We can also use the Monte Carlo simulation method that can help us to find the most likely prices of logs based on their values in the past. Nevertheless, the key concept of robust optimization is that we do not use probability distributions explicitly in our models, instead we can use them implicitly in the formulation of our uncertainty sets. I mentioned it before, but it is worth to repeat, that thanks to excluding explicit use of probability distributions in our model, we keep tractability of the optimization problem.

# From data to decisions

In the chart below, you can find simple flow, how data – small or big – can drive organizational success through fact-based, and proactive decision making, by the use of robust optimization technique in this case. Thanks to robust optimization we are getting optimal decision and some level of robustness.

# Pros and cons of Robust Optimization

I put the most important pros and cons of robust optimization in the table below. We should always be aware that the perfect method in the optimization does not exist, especially when models have many variables that can vary due to different levels of risks. The decision-makers and modelers should be always aware of the limitations of particular methods, and be able to select the appropriate one to the given problem. Some methods work better with certain problems, some not.

Mulvey et al. 1995, said in their article entitled “Robust Optimization of Large-Scale Systems”:

„Robust optimization is not a panacea for mathematical programming in the face of noisy data”

and we should keep his words in mind.

Table 1. Pros and cons of robust optimization.

Pros |
Cons |

No information on probability distribution needed | The big issue is to find the appropriate uncertainty set. This is an application dependent question. |

Only marginal complexity increase (compared to deterministic case) and remain linear!!! | Single solution without any flexibity! |

RO is tractable – easier to solve than the stochastic version | Very inprecise description of uncertainty even when feasible solutions exist, they may be over conservative, i.e., the optimal objective function value may be very high. |

RO formulations are more complex, and computationally more expensive, than their LP counterpart | |

Decision based on worst-case value over the uncertainty set. the size of the robust formulation grows with the number of scenarios and therefore this formulation becomes less tractable as the number of scenarios increases. |

# Summary

**Robust optimization techniques address the problem of data uncertainty in a particular optimization problem by optimizing the worst-case scenario. Robust optimization methods assume that the uncertainty set is known, with box-constrained set and ellipsoidal sets being the most common forms. Robust optimization has been successfully applied in various different fields such as engineering, finance, and statistics. Regarding the forest and wood industry sector, robust optimization has been often applied to timberland portfolio optimization, harvest scheduling models or forest management problems in general.**

**Forest Business Analytics (FBA) – a specialized consulting and business analytical company in forest and wood industry sectors – offer a wide range of optimization techniques applied to practical problems that your company faces every day. We know how to approach above problems, i.e. portfolio optimization, harvest scheduling models or forest management problems in general. FBA is changing business analytics forever by data preparation and analysis — giving unprecedented power to forest and wood industry products businesses without the need for cumbersome and expensive IT investments. Our multi-disciplinary team of experienced specialists investigates how data – small or big – can drive organizational success through fact-based, and proactive decision making. **

Author of the post:

Rafal Chudy – PhD Candidate in forest and resource economics at the Faculty of Environmental Sciences and Natural Resource Management, Norwegian University of Life Sciences (NMBU). He has acquired the international experience in forestry and forest economics at North Carolina State University, Swedish University of Agricultural Sciences, Oregon State University, University of Helsinki, University of Hamburg and Warsaw School of Economics. Rafal has gained profesional experience as forest economists and analyst at United States Department of Agriculture, National Forest Holding in Poland and many other companies from private sector.