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Online and stochastic optimization for the harvesting of short rotation coppice
Journal of Cleaner Production 110 (2016) 78e84
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Journal of Cleaner Production journal homepage: www.elsevier.com/locate/jclepro
Online and stochastic optimization for the harvesting of short rotation coppice Marco Bender a, Morten Tiedemann b, *, Laura Teuber c a
DFG RTG 1703, Institute for Applied Stochastics and Operations Research, Clausthal University of Technology, Erzstraße 1, D-38678 Clausthal-Zellerfeld, Germany b €ttingen, Lotzestraße 16-18, D-37083 Go €ttingen, Germany DFG RTG 1703, Institute for Numerical and Applied Mathematics, University of Go c €ttingen, Büsgenweg 4, D-37077 Go €ttingen, Germany DFG RTG 1703, Wood Biology and Wood Products, University of Go
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 July 2014 Received in revised form 30 August 2015 Accepted 31 August 2015 Available online 8 September 2015
Facing an increasing need of biomass and an expected gap between production of wood and utilization, wood from short rotation coppice offers great potential to cover the demand of renewable resources. Short rotation coppice is a form of agricultural land use to provide woodchips for energy supply or wood for industrial applications in a relatively short time. The economic profitability is the most important factor for the adoption of short rotation coppice for energy from biomass. Further on, the profitability of short rotation coppice was found to be most sensitive to the price for biomass and biomass yield. In this work, we present a simplified model for the problem of finding the optimal harvesting and selling policy with respect to a maximum profit by incorporating price and biomass yield as uncertain factors. Since we are dealing with renewable resources, the uncertain nature of these factors (e.g. due to insect infestation, annual weather conditions or unsteady prices) is taken into consideration and the model is especially designed to deal with these uncertainties. In order to incorporate the uncertainties inherent in the determining factors of the problem, the concepts of online optimization and stochastic optimization are applied to a simplified model of the problem. We then derive optimal policies and give explicit results. By means of these mathematical optimization techniques we can prove that our policies are best possible or produce results that are close to an optimal solution. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Online optimization Competitive analysis Stochastic optimization Short rotation coppice Economic profitability
1. Introduction The use of wood biomass for wood products and energy supply is increasingly gaining in importance, especially for meeting les-García regional demands of heat and electricity supply (Gonza et al., 2014). 1.1. Short rotation coppice Short rotation coppice (SRC) is a form of agricultural land use to provide woodchips for energy recovery or wood for industrial
* Corresponding author. E-mail addresses: [email protected] (M. Bender), m.tiedemann@ math.uni-goettingen.de (M. Tiedemann), [email protected] (L. Teuber). http://dx.doi.org/10.1016/j.jclepro.2015.08.120 0959-6526/© 2015 Elsevier Ltd. All rights reserved.
applications in a relatively short time (Knust et al., 2009). Hybrids of fast growing tree species with a high annual growth rate of biomass, e.g., poplar (Populus spp.) and willow (Salix spp.), are intensively cultivated via cuttage and stump shooting (cf. Knust et al., 2009; Ceulemans and Deraedt, 1999; Hauk et al., 2014). The trees are planted in rows at a space of 0.5e1.0 m and harvested manually or mechanically (depending on stem diameter) after a rotation period of one to 15 years. The profitability of SRC depends on maximizing the yield of biomass per unit area (Sage, 1999). The biomass yield depends on planting density, rotation time, and siterelated factors like soil condition, rainfall, competing vegetal overgrowth, and pests. To foster the biomass accumulation and enhance tree growth, land development measures can be applied, i.e., mechanical soil preparation, fertilization, irrigation, and weed control (cf. Knust et al., 2009; Ceulemans and Deraedt, 1999; Sage, 1999; Bilodeau-Gauthier et al., 2011).
M. Bender et al. / Journal of Cleaner Production 110 (2016) 78e84
Different types of wood exhibit varying growth properties, which determine among other things planting density and rotation times. While the cultivation of industrial wood requires a lower planting density and longer rotation times (10e15 years) (Knust et al., 2009), trees for energy recovery can be planted at a high density and at short rotation periods (usually 3 years for poplar). Furthermore, an increased rotation period leads to higher yields and better wood quality (increased wood/bark ratio) for each individual tree, but requires a lower planting density per unit area (Hauk et al., 2014). The economic profitability is singled out as the most important factor for the adoption of SRC for energy from biomass and the profitability of SRC was found to be most sensitive to the price for biomass and biomass yield. In this work, we model the problem of finding the optimal harvesting and selling policy with respect to a maximum profit by incorporating price and biomass yield as uncertain factors. Since we are dealing with renewable resources, the uncertain nature of these factors, e.g., due to pest infestation, annual weather conditions, or unsteady prices, is taken into consideration and the model is especially designed to deal with these uncertainties. 1.2. Motivation and limitations of the model We consider the problem of finding the optimal harvesting and selling policy by incorporating uncertain factors. In this work, this question is approached from a mathematical point of view, more precisely by means of online optimization (see Section 3) and by stochastic optimization (see Section 4). We follow the overall approach of applying optimization methods for problems with uncertainties that guarantee optimality of the solution (rather than heuristics whose solutions are in general not optimal). Within the field of optimization under uncertainty, the methods of online, stochastic, and robust optimization represent the most well-known approaches. Since robust optimization does not consider the dynamic aspect of the problem but rather all decisions are made in the beginning, we focus on online and stochastic optimization. The main goal of this work is to introduce these concepts to the field of renewable resources and to point out how these concepts may be of help to understand such problems in a better way. In order to achieve a model that is analyzable by means of these methods, we (over-)simplify the real-world problem and focus on the defining aspects of the problem. It is not our aspiration to reflect the real-world situation one-to-one in our model, but rather to focus on few main aspects and obtain an easily comprehensible model. Thus, the idea of how to apply the methods of online and stochastic optimization comes to the fore and we make a first step to applying these methods to more complex models. The cornerstones of our model are given as follows: We consider discrete time periods. In each time period, the amount of available wood, in the following referred to as resources, is increased by additional biomass yield and decreased by the amount sold by the manager. The selling options for the manager in each time period are determined by requests of potential buyers featuring a purchase quantity and an offered price the potential buyer is willing to pay for this quantity. Now, the question of the optimal harvesting and selling policy with respect to a maximum profit arises. As mentioned above, this is obviously a simplified model for the problem, which allows for a first analysis within the context of online and stochastic optimization. The model does not incorporate many aspects that certainly have an influence in the real-world problem, consider for example market dominance (“Is the market
79
entirely dominated by the buyers?”), variable biomass increase, time discounting on the revenue or cost streams, behavioral assumptions, etc. In the following, the problem of finding the optimal harvesting and selling policy is analyzed within the simplified model. We apply the concept of online optimization and present an optimal online algorithm. Furthermore, a stochastic optimization model for the problem is presented and explicit results are given, which demonstrate the power of additional stochastic information. 2. Online and stochastic optimization In classical combinatorial optimization, it is assumed that the input data of a problem instance is fully available. Based on this complete knowledge of data, an algorithm computes an optimal solution. In online optimization, an algorithm has to make a sequence of decisions, based on successively revealed information, that will have an impact on the final quality of its overall performance. Each of those decisions must be made without secure information about the future. Such an algorithm is called an online algorithm, in contrast to an offline algorithm that is aware of all relevant information in advance and computes a solution based on the full data set (cf. Borodin and El-Yaniv, 1998, Section1.1.1). Online algorithms are applied to decision problems that require decisions to be made immediately after new bits of information for the problem are revealed and no knowledge about future events is available. Considering the harvesting decision with respect to SRC, in each time period the decision about selling or waiting has to be made: The manager either harvests (parts of) the wood and sells it for the current price, or waits for a better price in the future (and a possible growth in biomass). However, there is no knowledge about future prices. The price of renewable resources, in this case SRC, is highly dependent on uncontrollable external influences such as weather conditions or pest infestations. The same holds true for the growth in biomass. Therefore, a worst-case analysis such as the concept of competitive analysis is appropriate in this case. In order to measure the quality of online algorithms, we use the well-known concept of competitive analysis, where, for each request sequence, the value obtained by an online algorithm is compared to the optimal value achievable on that sequence. We denote the value obtained by an online algorithm on a sequence s by ALG(s) and the optimal value achievable on that sequence by OPT(s). For an overview on the topic of competitive analysis for online problems we refer to the textbook by Borodin and El-Yaniv (1998). For maximization problems as studied in this paper, deterministic competitive algorithms are formally defined as stated in the following definition: Definition 2.1. (Deterministic Competitive Algorithm) A deterministic online algorithm ALG is called c-competitive for a constant c 1 if ALG(s) 1/c,OPT(s) for all request sequences s. The competitive ratio of a deterministic online algorithm is defined as the infimum over all c such that the algorithm is ccompetitive. Competitive analysis often yields very pessimistic results which is due to the worst-case nature of this analysis. For many practical problems, stochastic information about the unknown data is available. This can be incorporated in stochastic optimization (or stochastic programming), where piece by piece, some of the random variables are realized, and the task is to design a
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policy that makes at each of these points a decision that maximizes the sum of the current profits and the expected future profits given this choice. We assume that we are given probability distributions for the unknown bimoass price and yield. An introduction to stochastic optimization can be found in Shapiro et al. (2009).
3. Competitive analysis of the Online Harvesting Problem In this section, the problem described in Section 1.2 is formally introduced and then analyzed by means of competitive analysis. Consider a known time horizon T 2 ℕþ and n requests ri ¼ (di, pi, wi) (for i 2 {1,…,n}), each consisting of a time period di 2 {1,…,T} in which the request is offered, a price pi 2 ℝþ, and a purchase quantity wi 2 ℕþ. Thus, ri corresponds to a request of a potential buyer to buy wi units of resources at time di for a price pi. In each time period t 2 {1,…,T}, the requests ri with di ¼ t are revealed and an online algorithm has to decide which of these requests to accept. Note that, in each time period, multiple requests can be revealed. The requests with di ¼ t that are not accepted in time period t are lost. The resources are increased by kt 2 ℕþ in time period t. The resource increment kt in time period t is not known to the online player in advance. Denoting the available resources in time period t by ct, this means that c1 ¼ k1 and ct ¼ ct1 þ kt
P
wi for t 2, where It
1
denotes the set of indices of re-
i2It1
quests accepted by the online algorithm at time t 1. Observe that the biomass price and yield are incorporated as uncertainties in the model. The objective is to maximize the total value of accepted requests over all time periods 1,…, T while not accepting requests of total size larger than ct in any time period t. The problem described above is in the following referred to as the Online Harvesting Problem (OHP). Basically, the OHP is an online knapsack problem with increasing capacity. The requests presented to the online player in each time period featuring a price and a purchase quantity correspond to the items with a value and a weight which can be included in the knapsack. The available resources match the available knapsack capacity in each time period. The online knapsack problem with increasing capacity is investigated in Thielen et al. (2014). For the recapitulation of the results for the online knapsack problem with increasing capacity we use the terminology of the knapsack problem, i.e., items, weights, and values. The capacity increment is set to a constant k and competitive algorithms and lower bounds are deduced. For the case of unit weight requests and unit incremental capacity (i.e., one additional unit of capacity becoming available in each time period), a deterministic T-competitive online algorithm and a matching lower bound on the competitive ratio of any deterministic online algorithm is presented. For unit weights and k-incremental capacity (where k 2 additional units of capacity become available in each time period), a deterministic ((T þ 1)k/(2k 1))-competitive algorithm is given. For the case that general nonnegative weights are allowed, it is shown that no competitive online algorithm exists for the problem. However, for limited weights in {1,…, k} and k-incremental capacity, a deterministic online algorithm with a competitive ratio of
2T 1 and a lower bound on the competitive ratio of any deterministic online algorithm that approaches the competitive ratio of the proposed algorithm for k / ∞ is presented. Even for randomized algorithms the competitive ratios can at most be improved by a factor of 2. These results reveal the power of the adversary for this problem, leading to high competitive ratios depending linearly on the time horizon T. Therefore, we decrease the power of the adversary by slightly changing the problem setting. We assume that, in each time period, the online player is able to sell any amount of the available resource for a market price which is revealed to the online player in an online fashion. Thus, in each time period the resource increment kt and the current market price pt is presented to the online player and the decision of the online player is narrowed down to the amount of the available resources to be sold for the given price pt. Furthermore, pt is bounded, i.e., pmin pt pmax. The problem described above is in the following referred to as the Online Harvesting Problem with Bounded Market Prices (OHP with Bounded Market Prices). To the best of our knowledge, there is no previous work on the OHP with Bounded Market Prices as studied in this paper. The corresponding offline solution OPT to the OHP with Bounded Market Prices is given by selling each entire resource increment kt for the highest available market price, i.e.,
OPT ¼
T X t¼1
kt $
max
t 0 2ft;…;Tg
pt 0 :
(1)
In the remainder of this section, we present a deterministic algorithm for the OHP with Bounded Market Prices and prove that its competitive ratio is best possible. In Section 4, we return to the original OHP described in the beginning of this section and analyze this problem by stochastic optimization, which is another possibility to overcome the pessimism of competitive analysis. 3.1. A competitive algorithm For a fixed amount of resources (instead of an increasing amount of resources), the OHP with Bounded Market Prices is equivalent to the well-known one-way trading problem (El-Yaniv et al., 1992). Here, the online player is a trader whose goal is to trade some initial wealth given in some currency to some other currency. In each time period t, a new exchange rate pt is presented and the trader must decide on the fraction of the remaining wealth to be exchanged using the current exchange rate. The optimal deterministic algorithm THREAT for the classical one-way trading problem (with fixed initial wealth) is a threatbased policy, i.e., in each time period the trader exchanges just enough to ensure that the optimal competitive ratio would be obtained if the exchange rate is dropped to the minimum possible rate by the adversary for the rest of the game. We adapt this policy for the OHP with Bounded Market Prices and prove that the same competitive ratio as in the case with fixed initial wealth can be obtained. For this, we apply the threat-based policy to each available resource increment kt, which can be considered as a fixed wealth initiated at time t, see Algorithm 1.
M. Bender et al. / Journal of Cleaner Production 110 (2016) 78e84
Observe that pmin and pmax are the upper and lower bounds on the prices given in the definition of the model. The expressions pimax , i 2 {1,…,T} are used only in Algorithm 1. Theorem 3.1. Algorithm 1 is c+-competitive for the OHP with Bounded Market Prices, where c+ is the unique root of the equation
! pmin ðc 1Þ 1=T c ¼ T$ 1 : pmax pmin
c ¼ ðT t þ 1Þ$ 1
t2f1;…;Tg
(2)
Furthermore, no online algorithm for the OHP with Bounded Market Prices can obtain a smaller competitive ratio than c+. Proof. Consider an arbitrary t 2 {1,…,T} and the corresponding resource increment kt. Algorithm 1 sells a fraction st of this resource increment kt in time period t (see lines 6 and 9 of Algorithm 1), if the current price pt is the highest seen price so far for this resource increment (see lines 5 and 8 of Algorithm 1). Otherwise, the resource increment kt is not used. Note that Algorithm 1 acts only if the price is larger than c+,pmin because a competitive ratio of c+ is always attainable when the maximum rate is c+,pmin even if all of the resources are sold for the minimum price pmin. Consequently, we can assume that the price sequence with respect to the actions taken for resource kt is strictly increasing. Thus, we can identify the analysis of Algorithm 1 with respect to resource increment kt with the analysis of the classical one-way trading algorithm THREAT over a time horizon of T t þ 1, cf. (El-Yaniv et al., 1992). Here, the optimal fraction to trade is determined as 1/c+pt pminc+/pt pmin for the first (relevant) trading period, which corresponds to the fraction traded in line 6 in Algorithm 1, and as 1/c+pt pt 1/pt pmin which corresponds to the fraction traded in line 9 in Algorithm 1. Note that pt 1 is replaced by pimax, which is in our case the previous price in the strictly increasing price sequence for resource increment kt. Since c+ is the competitive ratio for THREAT with time horizon T, the profit of Algorithm 1 gained by means of resource increment kt, is c+ t -competitive with respect to the profit gained by the optimal offline algorithm by means of resource increment kt, where c+ t is the unique root of the equation
pmin ðc 1Þ pmax pmin
1 Ttþ1
! :
(3)
Consequently, the competitive ratio of Algorithm 1 is given by
max
81
+ + c+ t ¼ c1 ¼ c ;
(4)
since c+ t is strictly decreasing in t. This completes the analysis of the competitive ratio of Algorithm 1. For the special case of kt ¼ 0, t ¼ 2,…,T, the OHP with Bounded Market Prices coincides with the classical one-way trading problem. Therefore, the same lower bound as for the competitive ratio of any one-way trading algorithm holds for the competitive ratio of any algorithm for the OHP with bounded market prices. In El-Yaniv et al. (1992) it is shown, that no one-way trading algorithm can obtain a competitive ratio smaller than c+. This completes the proof. ▫ The following example illustrates the behavior of Algorithm 1 and the relation of the online and the offline solution. Example 3.1. Consider a time horizon of T ¼ 3, a lower bound pmin ¼ 100 and an upper bound pmax¼150. In the first time period, a market price p1 ¼ 130 and k1 ¼ 30 available resources are given. In order to determine the fraction of k1 sold by Algorithm 1, we have to calculate the competitive ratio c+, which results in c+ z 1.1224. Based on this data, Algorithm 1 computes
s1 ¼
1 p1 pmin c+ 1 130 100$1:1224 z0:5274; ¼ 1:1224 130 100 c+ p1 pmin
and sells s1,k1 z 15.8232 units of resources k1 for a price p1. In the second time period, a market price p2 ¼ 145 and k2 ¼ 40 additional resources are given. For resources k1, s2 is calculated as
s2 ¼
1 p2 p1max 1 145 130 z0:2969: ¼ 1:1224 145 100 c+ p2 pmin
Thus, Algorithm 1 sells s2,k1 z 8.1670 units of resources k1 for a price p2. For resources k2, s2 is calculated as
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s2 ¼
M. Bender et al. / Journal of Cleaner Production 110 (2016) 78e84
1 p2 pmin c+ 1 145 100,1:1224 z0:6486; ¼ 1:1224 145 100 c+ p2 pmin p2max
since ¼ 0. Thus, Algorithm 1 sells s2,k2 z 25.9444 units of resources k2 for a price p2. In the last period, t ¼ 3, a market price p3 ¼ 100 and k3 ¼ 25 is given. Since this is the last period, Algorithm 1 sells the remaining resources, i.e., 45.0654 units of resources, for a price p3. The profit of Algorithm 1 results in 11509.709. The optimal (offline) solution is given by 12650 (consider for each resource increment kt the maximal available price). Thus, the competitive ratio (for this instance) is given as 12650/11331.599 z 1.1163, whereas the theoretical (worst case) competitive ratio is given as c+ ¼ 1.1224. Within the framework of competitive analysis, Algorithm 1 is the best possible strategy for the OHP with Bounded Market Prices. Thus, we have found an optimal harvesting and selling policy. Obviously, this strategy tries to hedge against the worst case at any point in time and is thus not directly applicable to real-world situations. However, the straight application of policies is not the main goal of online optimization and competitive analysis, but rather the identification of problem specific crucial points. In this case we gained the insight that the better the knowledge of upper and lower bounds for the future market prices, the better the performance of the strategy. The optimal policy accepts a new price only if it is at least as good as the best price seen so far and if the price exceeds the threshold c+,pmin, which lies in between pmin and pmax. 3.2. Numerical results In order to show the practical relevance of Algorithm 1 (THREAT), we evaluate it on 10,000 randomly generated instances, for which we use lower and upper bounds on the prices (pmin ¼ 30, pmax ¼ 100) and increments (kmin ¼ 30, kmax ¼ 60) derived from the literature. We generate the variables using both a uniform distribution and a normal distribution (see Section 4.3 for the details). We compare THREAT with a second algorithm, a simple greedy strategy (that accepts in each period as much as possible), which we denote by GREEDY. The results of the ratios are summarized in Table 1. When compared with the optimal offline algorithm OPT, THREAT has in both settings a smaller average ratio and maximum ratio than GREEDY. Note that the competitive ratio proven in Theorem 3.1 is in our setting c+ z 1.432, and, in fact, the ratio of THREAT exceeds this value on none of the generated instances. 4. Stochastic optimization of the Online Harvesting Problem So far, we have analyzed the OHP from a competitive analysis point of view. We have seen that it produces, as a worst-case analysis, pessimistic results which are often not very meaningful, even for risk-averse decision makers. However, the worst-case instances that lead to these bad competitive ratios presented in the last section are somewhat artificial in the sense that they are very unlikely to occur.
4.1. A stochastic optimization model In this section, we want to apply stochastic optimization to overcome the pessimism of competitive analysis. Stochastic optimization is a widely used framework that allows to incorporate knowledge about probability distributions of the uncertain data. In our setting, we assume that we are given in advance the number of time periods T. However, for the requests that occur with their respective prices pi and purchase quantities wi, and the capacity increments kt for every period t we are only given probability distributions, and their values are revealed when the period begins. In our case, historical data could, e.g., be used to deduce probability distributions to model the uncertainties. Throughout this section we assume that we are given upper and lower bounds such that pmin pi pmax and wmin wi wmax for all requests i, and kmin kt kmax and nmin nt nmax for all time periods t. Our problem consists of a sequence of alternating decisions that have to be taken by the algorithm (which of the newly presented requests should be accepted?) and realizations of random variables which reveal some parts of the unknown data (all requests and the capacity increment for the current time period). The goal is to find a feasible policy that, for every period, chooses the requests such that the sum of the profits of the currently chosen requests and the expected profits in the following periods given this choice are maximized. Note that it could make sense to reject requests in the first period to save up some capacity, if we knew, e.g., that the expected number of requests and their profits in later periods are very high. We therefore use the following notation: For each period t 2 {1,…,T} a random variable xt is realized which reveals the requests I t ¼ I t(xt) (jI tj ¼ nt(xt)) that are presented, along with the prices pi(xt) and purchase quantities wi(xt) for all i 2 I t, and the capacity increment kt(xt). We assume that all uncertain data is integer and upper and lower bounds are known. Hence, the set of all possible scenarios for period t, which we denote by Xt , is finite. After the realization of xt, we have to make a decision on which of the requests that have been presented to us should be accepted, i.e., we have to choose some It 4 I t . However, we can only accept P requests such that their total purchase quantity wðIt Þ ¼ wi does i2It
not exceed the available capacity, which is given by the sum of the capacity increment kt and the capacity we saved up in the previous stages. If there are x units of capacity saved up from periods 1,…, t 1, and scenario xt occurs, we denote by F t(x, xt) the set of all feasible subsets of the requests that can be accepted, i.e., F t ðx; xt Þ ¼ fIt 4I t ðxt Þ : wðIt Þ x þ kt g. The problem of finding a feasible policy that maximizes the total expected profit can be summarized in the following stochastic program:
max
I1 2F 1 ðx1 Þ
pðI1 Þ þ Ex2 ½Q2 ðwðI1 Þ; x1 Þ;
(5)
where we have for k 2 {2,…, T 1}
Qk ðx; xk Þ ¼
max
pðIk Þ þ Exkþ1 ½Qkþ1 ðx þ wðIk Þ; xkþ1 Þ;
(6)
max
pðIT Þ
(7)
Ik 2F k ðx;xk Þ
and Table 1 Comparison of the ratios OPT/THREAT and OPT/GREEDY: “av” denotes the average of the ratio, “max” the maximum ratio, and “sd” the standard deviation for uniformly and normally distributed random variables. OPT/THREAT
Uniform Normal
QT ðx; xT Þ ¼
IT 2F T ðx;xT Þ
4.2. Solving the stochastic program
OPT/GREEDY
av
sd
max
av
sd
max
1.146 1.049
0.0977 0.02801
1.402 1.213
1.235 1.055
0.1503 0.03573
2.176 1.249
In order to determine an optimal policy, we can make use of the nested structure of the problem (5) and solve it with a backwards procedure.
M. Bender et al. / Journal of Cleaner Production 110 (2016) 78e84
We therefore start with the last stage problem (7), and solve QT(x,xT) for all possible values of capacities x saved up in the previous stages and scenarios xT 2 Xt . Note that this is possible since by assumption it holds that x2f0; 1; …; T$kmax g, and the set of scenarios XT is finite. Each of these problems is a KNAPSACK problem and can be solved in pseudo-polynomial time, e.g., using dynamic programming (cf. Martello and Toth, 1990). Next, assume that we have solved Qk þ 1(x, xk þ 1),…, QT(x,xT) for all possible values of x and scenarios xk þ 1,…, xT, and we wish to solve Qk(x,xk). First, note that the expected values of the profits in later periods (6) can easily be computed as we assume that we solved Qk þ 1(x, xk þ 1) before. We first neglect the expected value in (6), and simply solve the KNAPSACK problem
max
Ik 2F k ðx;xk Þ
pðIk Þ:
(8)
Besides the optimal solution, the dynamic program also yields for every l 2 {0,1,…, x} a maximum-profit set of requests Ikl 4I k ðxk Þ such that the total purchase quantity is at most l, i.e., wðIkl Þ l. These sets can thus be seen as the most profitable subsets of the requests using at most l capacity. In order to solve Qk(x, xk) for a fixed choice of x and xk, we can then choose over all l 2 {0,1,…, x} the set of requests Ikl that maximizes
p Ikl þ Exkþ1 ½Qkþ1 ðx þ l; xkþ1 Þ:
(9)
4.3. Numerical results In order to evaluate the performance of the stochastic program, we derive lower and upper bounds from the data given in Hauk et al. (2014). We fix a time horizon of T ¼ 5 since this resembles most of the data, and it allows for fast solvability of the algorithms. As there a only few data points in Hauk et al. (2014) that do not allow to derive meaningful probability distributions for our random variables, we distinguish two settings. First, we assume that all parameters are drawn uniformly at random between the upper and lower bounds given in Table 2 for every time period. Next, we assume that the random variables are sampled from a normal distribution, where the mean is chosen as the center of the intervals given by lower and upper bound, and the standard deviation is chosen such that there is only little mass outside the intervals (the probability that the sampled value is not contained in the interval is below 105). We compare the average profit of the solutions provided by stochastic optimization (SP) with the average profits of: OPT e the optimal offline algorithm knows all increments and requests in advance, and GREEDY e the greedy algorithm acts in every period such that it maximizes the current profit using the available requests. OPT can be determined by solving an integer program as shown in Thielen et al. (2014). Additionally, the algorithm GREEDY was
proposed and turned out to be an algorithm with best-possible competitive ratio for many settings. Observe that we do not consider Algorithm 1 in this section since it is only applicable to a restricted setting (where the online player can sell any amount on the market). As the number of possible combinations of scenarios is too high for a full enumeration, we sample random instances (that consist of one scenario per time period) and use the average profit as an estimate for the expected profit. For a sample size of 10,000 instances we obtain the results presented in the rows with a ¼ 0 in Table 3. Obviously, OPT yields the highest expected profit. The average performance of GREEDY is better than its worst-case performance which only allowed poor competitive ratios (cf. Section 3). However, SP outperforms GREEDY. The better performance of SP follows from the fact that GREEDY does not make any use of the given probability distributions, and SP is tailor-made to guarantee a bestpossible average performance. The advantage of SP becomes even stronger when the probability distributions of the unknown data differ from period to period. This can be seen in the rows for a ¼ 0.2 and a ¼ 1 in Table 3: We modify the scenarios in such a way that all probability distributions remain unchanged, except pmin and pmax are increased in every period by 20% and 100%, respectively, i.e., pmin(t þ 1) ¼ (1 þ a),pmin(t) and pmax(t þ 1) ¼ (1 þ a),pmax(t) for a 2 {0.2,1}. The SP solution tends to save up capacity in the first stages (as it uses the fact that very likely more profitable requests will appear later on), whereas GREEDY simply aims at maximizing the profit in every period. To make our statements more precise, we also performed a onesided t-test for the hypothesis that the objective value of GREEDY exceeds the value of SP. For all settings presented above, we obtained p-values below 105, i.e., the hypothesis can be rejected. The generation of the instances and the dynamic program were implemented in Cþþ using as compiler GCC 4.9.2, and the integer programs for OPT were solved using IBM ILOG CPLEX Optimization Studio 12.5. 5. Conclusion and future work In this work, the economic profitability of short rotation coppice is approached from the point of view of mathematical optimization. In order to incorporate the uncertainties inherent in the determining factors of the problem, i.e., price and biomass yield, the concepts of online optimization and stochastic optimization are applied to the problem. Online algorithms and a stochastic optimization model are proposed, giving possible policies to approach the problem and further insight into the nature of the problem. While previous work on online optimization shows that GREEDY strategies perform well, this is not the end of the story. For a setting with bounded market prices, which has not been researched so far, we present a more sophisticated algorithm THREAT, which we prove to be best possible from the viewpoint of competitive Table 3 Comparison of the average profits (av) and standard deviations (sd) of the solutions obtained by stochastic programming (SP), greedy algorithm (GREEDY), and the optimal offline algorithm (OPT) with price increase factor a 2 {0,0.2,1}.
Table 2 Lower and upper bounds for the evaluation of the stochastic programming solution.
k n p w
min
max
30 1 30 1
60 10 100 100
83
SP
Uniform
Normal
a¼0 a ¼ 0.2 a¼1 a¼0 a ¼ 0.2 a¼1
GREEDY
OPT
av
sd
av
sd
av
sd
625.8 940.6 5096.8 287.5 537.2 4101.9
55.4 51.9 103.1 5.9 7.1 65.1
604.7 863.1 3943.3 281.2 395.2 1764.4
143.9 219.9 1309.6 30.1 57.8 401.8
635.6 984.5 5485.9 293.9 541.0 4123.7
139.2 231.1 1646.8 11.1 22.4 197.4
84
M. Bender et al. / Journal of Cleaner Production 110 (2016) 78e84
analysis. Besides this theoretical result, our simulations indicate that it outperforms GREEDY on randomly generated instances. For the case that we are given probability distributions for the unknown variables, we show how stochastic optimization can be used to determine an optimal policy (with respect to the expected value). Again, our numerical results show that this approach is superior to a simple GREEDY strategy. Both of these findings show that it is worthwhile to design tailor-made algorithms which find optimal (in a certain sense) solutions, rather than simply applying some straightforward heuristic. Future work includes incorporating further aspects of the problem into the mathematical formulation and trying to achieve models which are closer to the real-world problem. For example, it might be interesting to study correlated random variables (in practical application, biomass price and yield are most likely correlated) or discounting effects. While some of the present algorithms are applicable to more general settings, it is not clear how to analyze their performance theoretically or how to design algorithms that take the additional information into account and yield better results. Acknowledgments This research was supported by the German Research Foundation (DFG), grant GRK 1703/1 for the Research Training Group “Resource Efficiency in Interorganizational Networks e Planning Methods to Utilize Renewable Resources”.
References , D., Messier, C., Be langer, N., 2011. Juvenile growth of Bilodeau-Gauthier, S., Pare hybrid poplars on acidic boreal soil determined by environmental effects of soil preparation, vegetation control, and fertilization. For. Ecol. Manag. 261, 620e629. Borodin, A., El-Yaniv, R., 1998. Online Computation and Competitive Analysis. Cambridge University Press. Ceulemans, R., Deraedt, W., 1999. Production physiology and growth potential of poplars under short-rotation forestry culture. For. Ecol. Manag. 121, 9e23. El-Yaniv, R., Turpin, G., Karp, R., Fiat, A., 1992. Competitive analysis of financial games. In: Proceedings of the 33rd Annual IEEE Symposium on the Foundations of Computer Science (FOCS), pp. 327e333. Gonz ales-García, S., Moreira, M.T., Dias, A.C., Mola-Yudego, B., 2014. Cradle-to-gate life cycle assessment of forest operations in Europe: environmental and energy profiles. J. Clean. Prod. 66, 188e198. Hauk, S., Knoke, T., Wittkopf, S., 2014. Economic evaluation of short rotation coppice systems for energy from biomass e a review. Renew. Sustain. Energy Rev. 29, 435e448. Knust, C., 2009. Kurzumtriebsplantagen e stand des wissens. In: Reeg, T., Bemmann, A., Konold, W., Murach, D., Spiecker, H. (Eds.), Anbau und Nutzung €umen auf landwirtschaftlichen Fl€ von Ba achen. WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, pp. 3e9. Martello, S., Toth, P., 1990. Knapsack Problems: Algorithms and Computer Implementations. John Wiley & Sons, Chichester. Sage, R., 1999. Weed competition in willow coppice crops: the cause and extent of yield losses. Weed Res. 39 (5), 399e311. Shapiro, A., Dentcheva, D., Ruszczynski, A., 2009. Lectures on Stochastic Programming: Modeling and Theory, MPS-SIAM Series on Optimization, Society for Industrial and Applied Mathematics, Philadelphia. http://opac.inria.fr/ record¼b1130095. Thielen, C., Tiedemann, M., Westphal, S., 2014. The Online Knapsack Problem with Incremental Capacity. Tech. Rep. 2014-02. Preprint-Reihe, Institut für Numerische und Angewandte Mathematik, Georg-August Universit€ at, €ttingen. Go
Contents lists available at ScienceDirect
Journal of Cleaner Production journal homepage: www.elsevier.com/locate/jclepro
Online and stochastic optimization for the harvesting of short rotation coppice Marco Bender a, Morten Tiedemann b, *, Laura Teuber c a
DFG RTG 1703, Institute for Applied Stochastics and Operations Research, Clausthal University of Technology, Erzstraße 1, D-38678 Clausthal-Zellerfeld, Germany b €ttingen, Lotzestraße 16-18, D-37083 Go €ttingen, Germany DFG RTG 1703, Institute for Numerical and Applied Mathematics, University of Go c €ttingen, Büsgenweg 4, D-37077 Go €ttingen, Germany DFG RTG 1703, Wood Biology and Wood Products, University of Go
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 July 2014 Received in revised form 30 August 2015 Accepted 31 August 2015 Available online 8 September 2015
Facing an increasing need of biomass and an expected gap between production of wood and utilization, wood from short rotation coppice offers great potential to cover the demand of renewable resources. Short rotation coppice is a form of agricultural land use to provide woodchips for energy supply or wood for industrial applications in a relatively short time. The economic profitability is the most important factor for the adoption of short rotation coppice for energy from biomass. Further on, the profitability of short rotation coppice was found to be most sensitive to the price for biomass and biomass yield. In this work, we present a simplified model for the problem of finding the optimal harvesting and selling policy with respect to a maximum profit by incorporating price and biomass yield as uncertain factors. Since we are dealing with renewable resources, the uncertain nature of these factors (e.g. due to insect infestation, annual weather conditions or unsteady prices) is taken into consideration and the model is especially designed to deal with these uncertainties. In order to incorporate the uncertainties inherent in the determining factors of the problem, the concepts of online optimization and stochastic optimization are applied to a simplified model of the problem. We then derive optimal policies and give explicit results. By means of these mathematical optimization techniques we can prove that our policies are best possible or produce results that are close to an optimal solution. © 2015 Elsevier Ltd. All rights reserved.
Keywords: Online optimization Competitive analysis Stochastic optimization Short rotation coppice Economic profitability
1. Introduction The use of wood biomass for wood products and energy supply is increasingly gaining in importance, especially for meeting les-García regional demands of heat and electricity supply (Gonza et al., 2014). 1.1. Short rotation coppice Short rotation coppice (SRC) is a form of agricultural land use to provide woodchips for energy recovery or wood for industrial
* Corresponding author. E-mail addresses: [email protected] (M. Bender), m.tiedemann@ math.uni-goettingen.de (M. Tiedemann), [email protected] (L. Teuber). http://dx.doi.org/10.1016/j.jclepro.2015.08.120 0959-6526/© 2015 Elsevier Ltd. All rights reserved.
applications in a relatively short time (Knust et al., 2009). Hybrids of fast growing tree species with a high annual growth rate of biomass, e.g., poplar (Populus spp.) and willow (Salix spp.), are intensively cultivated via cuttage and stump shooting (cf. Knust et al., 2009; Ceulemans and Deraedt, 1999; Hauk et al., 2014). The trees are planted in rows at a space of 0.5e1.0 m and harvested manually or mechanically (depending on stem diameter) after a rotation period of one to 15 years. The profitability of SRC depends on maximizing the yield of biomass per unit area (Sage, 1999). The biomass yield depends on planting density, rotation time, and siterelated factors like soil condition, rainfall, competing vegetal overgrowth, and pests. To foster the biomass accumulation and enhance tree growth, land development measures can be applied, i.e., mechanical soil preparation, fertilization, irrigation, and weed control (cf. Knust et al., 2009; Ceulemans and Deraedt, 1999; Sage, 1999; Bilodeau-Gauthier et al., 2011).
M. Bender et al. / Journal of Cleaner Production 110 (2016) 78e84
Different types of wood exhibit varying growth properties, which determine among other things planting density and rotation times. While the cultivation of industrial wood requires a lower planting density and longer rotation times (10e15 years) (Knust et al., 2009), trees for energy recovery can be planted at a high density and at short rotation periods (usually 3 years for poplar). Furthermore, an increased rotation period leads to higher yields and better wood quality (increased wood/bark ratio) for each individual tree, but requires a lower planting density per unit area (Hauk et al., 2014). The economic profitability is singled out as the most important factor for the adoption of SRC for energy from biomass and the profitability of SRC was found to be most sensitive to the price for biomass and biomass yield. In this work, we model the problem of finding the optimal harvesting and selling policy with respect to a maximum profit by incorporating price and biomass yield as uncertain factors. Since we are dealing with renewable resources, the uncertain nature of these factors, e.g., due to pest infestation, annual weather conditions, or unsteady prices, is taken into consideration and the model is especially designed to deal with these uncertainties. 1.2. Motivation and limitations of the model We consider the problem of finding the optimal harvesting and selling policy by incorporating uncertain factors. In this work, this question is approached from a mathematical point of view, more precisely by means of online optimization (see Section 3) and by stochastic optimization (see Section 4). We follow the overall approach of applying optimization methods for problems with uncertainties that guarantee optimality of the solution (rather than heuristics whose solutions are in general not optimal). Within the field of optimization under uncertainty, the methods of online, stochastic, and robust optimization represent the most well-known approaches. Since robust optimization does not consider the dynamic aspect of the problem but rather all decisions are made in the beginning, we focus on online and stochastic optimization. The main goal of this work is to introduce these concepts to the field of renewable resources and to point out how these concepts may be of help to understand such problems in a better way. In order to achieve a model that is analyzable by means of these methods, we (over-)simplify the real-world problem and focus on the defining aspects of the problem. It is not our aspiration to reflect the real-world situation one-to-one in our model, but rather to focus on few main aspects and obtain an easily comprehensible model. Thus, the idea of how to apply the methods of online and stochastic optimization comes to the fore and we make a first step to applying these methods to more complex models. The cornerstones of our model are given as follows: We consider discrete time periods. In each time period, the amount of available wood, in the following referred to as resources, is increased by additional biomass yield and decreased by the amount sold by the manager. The selling options for the manager in each time period are determined by requests of potential buyers featuring a purchase quantity and an offered price the potential buyer is willing to pay for this quantity. Now, the question of the optimal harvesting and selling policy with respect to a maximum profit arises. As mentioned above, this is obviously a simplified model for the problem, which allows for a first analysis within the context of online and stochastic optimization. The model does not incorporate many aspects that certainly have an influence in the real-world problem, consider for example market dominance (“Is the market
79
entirely dominated by the buyers?”), variable biomass increase, time discounting on the revenue or cost streams, behavioral assumptions, etc. In the following, the problem of finding the optimal harvesting and selling policy is analyzed within the simplified model. We apply the concept of online optimization and present an optimal online algorithm. Furthermore, a stochastic optimization model for the problem is presented and explicit results are given, which demonstrate the power of additional stochastic information. 2. Online and stochastic optimization In classical combinatorial optimization, it is assumed that the input data of a problem instance is fully available. Based on this complete knowledge of data, an algorithm computes an optimal solution. In online optimization, an algorithm has to make a sequence of decisions, based on successively revealed information, that will have an impact on the final quality of its overall performance. Each of those decisions must be made without secure information about the future. Such an algorithm is called an online algorithm, in contrast to an offline algorithm that is aware of all relevant information in advance and computes a solution based on the full data set (cf. Borodin and El-Yaniv, 1998, Section1.1.1). Online algorithms are applied to decision problems that require decisions to be made immediately after new bits of information for the problem are revealed and no knowledge about future events is available. Considering the harvesting decision with respect to SRC, in each time period the decision about selling or waiting has to be made: The manager either harvests (parts of) the wood and sells it for the current price, or waits for a better price in the future (and a possible growth in biomass). However, there is no knowledge about future prices. The price of renewable resources, in this case SRC, is highly dependent on uncontrollable external influences such as weather conditions or pest infestations. The same holds true for the growth in biomass. Therefore, a worst-case analysis such as the concept of competitive analysis is appropriate in this case. In order to measure the quality of online algorithms, we use the well-known concept of competitive analysis, where, for each request sequence, the value obtained by an online algorithm is compared to the optimal value achievable on that sequence. We denote the value obtained by an online algorithm on a sequence s by ALG(s) and the optimal value achievable on that sequence by OPT(s). For an overview on the topic of competitive analysis for online problems we refer to the textbook by Borodin and El-Yaniv (1998). For maximization problems as studied in this paper, deterministic competitive algorithms are formally defined as stated in the following definition: Definition 2.1. (Deterministic Competitive Algorithm) A deterministic online algorithm ALG is called c-competitive for a constant c 1 if ALG(s) 1/c,OPT(s) for all request sequences s. The competitive ratio of a deterministic online algorithm is defined as the infimum over all c such that the algorithm is ccompetitive. Competitive analysis often yields very pessimistic results which is due to the worst-case nature of this analysis. For many practical problems, stochastic information about the unknown data is available. This can be incorporated in stochastic optimization (or stochastic programming), where piece by piece, some of the random variables are realized, and the task is to design a
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policy that makes at each of these points a decision that maximizes the sum of the current profits and the expected future profits given this choice. We assume that we are given probability distributions for the unknown bimoass price and yield. An introduction to stochastic optimization can be found in Shapiro et al. (2009).
3. Competitive analysis of the Online Harvesting Problem In this section, the problem described in Section 1.2 is formally introduced and then analyzed by means of competitive analysis. Consider a known time horizon T 2 ℕþ and n requests ri ¼ (di, pi, wi) (for i 2 {1,…,n}), each consisting of a time period di 2 {1,…,T} in which the request is offered, a price pi 2 ℝþ, and a purchase quantity wi 2 ℕþ. Thus, ri corresponds to a request of a potential buyer to buy wi units of resources at time di for a price pi. In each time period t 2 {1,…,T}, the requests ri with di ¼ t are revealed and an online algorithm has to decide which of these requests to accept. Note that, in each time period, multiple requests can be revealed. The requests with di ¼ t that are not accepted in time period t are lost. The resources are increased by kt 2 ℕþ in time period t. The resource increment kt in time period t is not known to the online player in advance. Denoting the available resources in time period t by ct, this means that c1 ¼ k1 and ct ¼ ct1 þ kt
P
wi for t 2, where It
1
denotes the set of indices of re-
i2It1
quests accepted by the online algorithm at time t 1. Observe that the biomass price and yield are incorporated as uncertainties in the model. The objective is to maximize the total value of accepted requests over all time periods 1,…, T while not accepting requests of total size larger than ct in any time period t. The problem described above is in the following referred to as the Online Harvesting Problem (OHP). Basically, the OHP is an online knapsack problem with increasing capacity. The requests presented to the online player in each time period featuring a price and a purchase quantity correspond to the items with a value and a weight which can be included in the knapsack. The available resources match the available knapsack capacity in each time period. The online knapsack problem with increasing capacity is investigated in Thielen et al. (2014). For the recapitulation of the results for the online knapsack problem with increasing capacity we use the terminology of the knapsack problem, i.e., items, weights, and values. The capacity increment is set to a constant k and competitive algorithms and lower bounds are deduced. For the case of unit weight requests and unit incremental capacity (i.e., one additional unit of capacity becoming available in each time period), a deterministic T-competitive online algorithm and a matching lower bound on the competitive ratio of any deterministic online algorithm is presented. For unit weights and k-incremental capacity (where k 2 additional units of capacity become available in each time period), a deterministic ((T þ 1)k/(2k 1))-competitive algorithm is given. For the case that general nonnegative weights are allowed, it is shown that no competitive online algorithm exists for the problem. However, for limited weights in {1,…, k} and k-incremental capacity, a deterministic online algorithm with a competitive ratio of
2T 1 and a lower bound on the competitive ratio of any deterministic online algorithm that approaches the competitive ratio of the proposed algorithm for k / ∞ is presented. Even for randomized algorithms the competitive ratios can at most be improved by a factor of 2. These results reveal the power of the adversary for this problem, leading to high competitive ratios depending linearly on the time horizon T. Therefore, we decrease the power of the adversary by slightly changing the problem setting. We assume that, in each time period, the online player is able to sell any amount of the available resource for a market price which is revealed to the online player in an online fashion. Thus, in each time period the resource increment kt and the current market price pt is presented to the online player and the decision of the online player is narrowed down to the amount of the available resources to be sold for the given price pt. Furthermore, pt is bounded, i.e., pmin pt pmax. The problem described above is in the following referred to as the Online Harvesting Problem with Bounded Market Prices (OHP with Bounded Market Prices). To the best of our knowledge, there is no previous work on the OHP with Bounded Market Prices as studied in this paper. The corresponding offline solution OPT to the OHP with Bounded Market Prices is given by selling each entire resource increment kt for the highest available market price, i.e.,
OPT ¼
T X t¼1
kt $
max
t 0 2ft;…;Tg
pt 0 :
(1)
In the remainder of this section, we present a deterministic algorithm for the OHP with Bounded Market Prices and prove that its competitive ratio is best possible. In Section 4, we return to the original OHP described in the beginning of this section and analyze this problem by stochastic optimization, which is another possibility to overcome the pessimism of competitive analysis. 3.1. A competitive algorithm For a fixed amount of resources (instead of an increasing amount of resources), the OHP with Bounded Market Prices is equivalent to the well-known one-way trading problem (El-Yaniv et al., 1992). Here, the online player is a trader whose goal is to trade some initial wealth given in some currency to some other currency. In each time period t, a new exchange rate pt is presented and the trader must decide on the fraction of the remaining wealth to be exchanged using the current exchange rate. The optimal deterministic algorithm THREAT for the classical one-way trading problem (with fixed initial wealth) is a threatbased policy, i.e., in each time period the trader exchanges just enough to ensure that the optimal competitive ratio would be obtained if the exchange rate is dropped to the minimum possible rate by the adversary for the rest of the game. We adapt this policy for the OHP with Bounded Market Prices and prove that the same competitive ratio as in the case with fixed initial wealth can be obtained. For this, we apply the threat-based policy to each available resource increment kt, which can be considered as a fixed wealth initiated at time t, see Algorithm 1.
M. Bender et al. / Journal of Cleaner Production 110 (2016) 78e84
Observe that pmin and pmax are the upper and lower bounds on the prices given in the definition of the model. The expressions pimax , i 2 {1,…,T} are used only in Algorithm 1. Theorem 3.1. Algorithm 1 is c+-competitive for the OHP with Bounded Market Prices, where c+ is the unique root of the equation
! pmin ðc 1Þ 1=T c ¼ T$ 1 : pmax pmin
c ¼ ðT t þ 1Þ$ 1
t2f1;…;Tg
(2)
Furthermore, no online algorithm for the OHP with Bounded Market Prices can obtain a smaller competitive ratio than c+. Proof. Consider an arbitrary t 2 {1,…,T} and the corresponding resource increment kt. Algorithm 1 sells a fraction st of this resource increment kt in time period t (see lines 6 and 9 of Algorithm 1), if the current price pt is the highest seen price so far for this resource increment (see lines 5 and 8 of Algorithm 1). Otherwise, the resource increment kt is not used. Note that Algorithm 1 acts only if the price is larger than c+,pmin because a competitive ratio of c+ is always attainable when the maximum rate is c+,pmin even if all of the resources are sold for the minimum price pmin. Consequently, we can assume that the price sequence with respect to the actions taken for resource kt is strictly increasing. Thus, we can identify the analysis of Algorithm 1 with respect to resource increment kt with the analysis of the classical one-way trading algorithm THREAT over a time horizon of T t þ 1, cf. (El-Yaniv et al., 1992). Here, the optimal fraction to trade is determined as 1/c+pt pminc+/pt pmin for the first (relevant) trading period, which corresponds to the fraction traded in line 6 in Algorithm 1, and as 1/c+pt pt 1/pt pmin which corresponds to the fraction traded in line 9 in Algorithm 1. Note that pt 1 is replaced by pimax, which is in our case the previous price in the strictly increasing price sequence for resource increment kt. Since c+ is the competitive ratio for THREAT with time horizon T, the profit of Algorithm 1 gained by means of resource increment kt, is c+ t -competitive with respect to the profit gained by the optimal offline algorithm by means of resource increment kt, where c+ t is the unique root of the equation
pmin ðc 1Þ pmax pmin
1 Ttþ1
! :
(3)
Consequently, the competitive ratio of Algorithm 1 is given by
max
81
+ + c+ t ¼ c1 ¼ c ;
(4)
since c+ t is strictly decreasing in t. This completes the analysis of the competitive ratio of Algorithm 1. For the special case of kt ¼ 0, t ¼ 2,…,T, the OHP with Bounded Market Prices coincides with the classical one-way trading problem. Therefore, the same lower bound as for the competitive ratio of any one-way trading algorithm holds for the competitive ratio of any algorithm for the OHP with bounded market prices. In El-Yaniv et al. (1992) it is shown, that no one-way trading algorithm can obtain a competitive ratio smaller than c+. This completes the proof. ▫ The following example illustrates the behavior of Algorithm 1 and the relation of the online and the offline solution. Example 3.1. Consider a time horizon of T ¼ 3, a lower bound pmin ¼ 100 and an upper bound pmax¼150. In the first time period, a market price p1 ¼ 130 and k1 ¼ 30 available resources are given. In order to determine the fraction of k1 sold by Algorithm 1, we have to calculate the competitive ratio c+, which results in c+ z 1.1224. Based on this data, Algorithm 1 computes
s1 ¼
1 p1 pmin c+ 1 130 100$1:1224 z0:5274; ¼ 1:1224 130 100 c+ p1 pmin
and sells s1,k1 z 15.8232 units of resources k1 for a price p1. In the second time period, a market price p2 ¼ 145 and k2 ¼ 40 additional resources are given. For resources k1, s2 is calculated as
s2 ¼
1 p2 p1max 1 145 130 z0:2969: ¼ 1:1224 145 100 c+ p2 pmin
Thus, Algorithm 1 sells s2,k1 z 8.1670 units of resources k1 for a price p2. For resources k2, s2 is calculated as
82
s2 ¼
M. Bender et al. / Journal of Cleaner Production 110 (2016) 78e84
1 p2 pmin c+ 1 145 100,1:1224 z0:6486; ¼ 1:1224 145 100 c+ p2 pmin p2max
since ¼ 0. Thus, Algorithm 1 sells s2,k2 z 25.9444 units of resources k2 for a price p2. In the last period, t ¼ 3, a market price p3 ¼ 100 and k3 ¼ 25 is given. Since this is the last period, Algorithm 1 sells the remaining resources, i.e., 45.0654 units of resources, for a price p3. The profit of Algorithm 1 results in 11509.709. The optimal (offline) solution is given by 12650 (consider for each resource increment kt the maximal available price). Thus, the competitive ratio (for this instance) is given as 12650/11331.599 z 1.1163, whereas the theoretical (worst case) competitive ratio is given as c+ ¼ 1.1224. Within the framework of competitive analysis, Algorithm 1 is the best possible strategy for the OHP with Bounded Market Prices. Thus, we have found an optimal harvesting and selling policy. Obviously, this strategy tries to hedge against the worst case at any point in time and is thus not directly applicable to real-world situations. However, the straight application of policies is not the main goal of online optimization and competitive analysis, but rather the identification of problem specific crucial points. In this case we gained the insight that the better the knowledge of upper and lower bounds for the future market prices, the better the performance of the strategy. The optimal policy accepts a new price only if it is at least as good as the best price seen so far and if the price exceeds the threshold c+,pmin, which lies in between pmin and pmax. 3.2. Numerical results In order to show the practical relevance of Algorithm 1 (THREAT), we evaluate it on 10,000 randomly generated instances, for which we use lower and upper bounds on the prices (pmin ¼ 30, pmax ¼ 100) and increments (kmin ¼ 30, kmax ¼ 60) derived from the literature. We generate the variables using both a uniform distribution and a normal distribution (see Section 4.3 for the details). We compare THREAT with a second algorithm, a simple greedy strategy (that accepts in each period as much as possible), which we denote by GREEDY. The results of the ratios are summarized in Table 1. When compared with the optimal offline algorithm OPT, THREAT has in both settings a smaller average ratio and maximum ratio than GREEDY. Note that the competitive ratio proven in Theorem 3.1 is in our setting c+ z 1.432, and, in fact, the ratio of THREAT exceeds this value on none of the generated instances. 4. Stochastic optimization of the Online Harvesting Problem So far, we have analyzed the OHP from a competitive analysis point of view. We have seen that it produces, as a worst-case analysis, pessimistic results which are often not very meaningful, even for risk-averse decision makers. However, the worst-case instances that lead to these bad competitive ratios presented in the last section are somewhat artificial in the sense that they are very unlikely to occur.
4.1. A stochastic optimization model In this section, we want to apply stochastic optimization to overcome the pessimism of competitive analysis. Stochastic optimization is a widely used framework that allows to incorporate knowledge about probability distributions of the uncertain data. In our setting, we assume that we are given in advance the number of time periods T. However, for the requests that occur with their respective prices pi and purchase quantities wi, and the capacity increments kt for every period t we are only given probability distributions, and their values are revealed when the period begins. In our case, historical data could, e.g., be used to deduce probability distributions to model the uncertainties. Throughout this section we assume that we are given upper and lower bounds such that pmin pi pmax and wmin wi wmax for all requests i, and kmin kt kmax and nmin nt nmax for all time periods t. Our problem consists of a sequence of alternating decisions that have to be taken by the algorithm (which of the newly presented requests should be accepted?) and realizations of random variables which reveal some parts of the unknown data (all requests and the capacity increment for the current time period). The goal is to find a feasible policy that, for every period, chooses the requests such that the sum of the profits of the currently chosen requests and the expected profits in the following periods given this choice are maximized. Note that it could make sense to reject requests in the first period to save up some capacity, if we knew, e.g., that the expected number of requests and their profits in later periods are very high. We therefore use the following notation: For each period t 2 {1,…,T} a random variable xt is realized which reveals the requests I t ¼ I t(xt) (jI tj ¼ nt(xt)) that are presented, along with the prices pi(xt) and purchase quantities wi(xt) for all i 2 I t, and the capacity increment kt(xt). We assume that all uncertain data is integer and upper and lower bounds are known. Hence, the set of all possible scenarios for period t, which we denote by Xt , is finite. After the realization of xt, we have to make a decision on which of the requests that have been presented to us should be accepted, i.e., we have to choose some It 4 I t . However, we can only accept P requests such that their total purchase quantity wðIt Þ ¼ wi does i2It
not exceed the available capacity, which is given by the sum of the capacity increment kt and the capacity we saved up in the previous stages. If there are x units of capacity saved up from periods 1,…, t 1, and scenario xt occurs, we denote by F t(x, xt) the set of all feasible subsets of the requests that can be accepted, i.e., F t ðx; xt Þ ¼ fIt 4I t ðxt Þ : wðIt Þ x þ kt g. The problem of finding a feasible policy that maximizes the total expected profit can be summarized in the following stochastic program:
max
I1 2F 1 ðx1 Þ
pðI1 Þ þ Ex2 ½Q2 ðwðI1 Þ; x1 Þ;
(5)
where we have for k 2 {2,…, T 1}
Qk ðx; xk Þ ¼
max
pðIk Þ þ Exkþ1 ½Qkþ1 ðx þ wðIk Þ; xkþ1 Þ;
(6)
max
pðIT Þ
(7)
Ik 2F k ðx;xk Þ
and Table 1 Comparison of the ratios OPT/THREAT and OPT/GREEDY: “av” denotes the average of the ratio, “max” the maximum ratio, and “sd” the standard deviation for uniformly and normally distributed random variables. OPT/THREAT
Uniform Normal
QT ðx; xT Þ ¼
IT 2F T ðx;xT Þ
4.2. Solving the stochastic program
OPT/GREEDY
av
sd
max
av
sd
max
1.146 1.049
0.0977 0.02801
1.402 1.213
1.235 1.055
0.1503 0.03573
2.176 1.249
In order to determine an optimal policy, we can make use of the nested structure of the problem (5) and solve it with a backwards procedure.
M. Bender et al. / Journal of Cleaner Production 110 (2016) 78e84
We therefore start with the last stage problem (7), and solve QT(x,xT) for all possible values of capacities x saved up in the previous stages and scenarios xT 2 Xt . Note that this is possible since by assumption it holds that x2f0; 1; …; T$kmax g, and the set of scenarios XT is finite. Each of these problems is a KNAPSACK problem and can be solved in pseudo-polynomial time, e.g., using dynamic programming (cf. Martello and Toth, 1990). Next, assume that we have solved Qk þ 1(x, xk þ 1),…, QT(x,xT) for all possible values of x and scenarios xk þ 1,…, xT, and we wish to solve Qk(x,xk). First, note that the expected values of the profits in later periods (6) can easily be computed as we assume that we solved Qk þ 1(x, xk þ 1) before. We first neglect the expected value in (6), and simply solve the KNAPSACK problem
max
Ik 2F k ðx;xk Þ
pðIk Þ:
(8)
Besides the optimal solution, the dynamic program also yields for every l 2 {0,1,…, x} a maximum-profit set of requests Ikl 4I k ðxk Þ such that the total purchase quantity is at most l, i.e., wðIkl Þ l. These sets can thus be seen as the most profitable subsets of the requests using at most l capacity. In order to solve Qk(x, xk) for a fixed choice of x and xk, we can then choose over all l 2 {0,1,…, x} the set of requests Ikl that maximizes
p Ikl þ Exkþ1 ½Qkþ1 ðx þ l; xkþ1 Þ:
(9)
4.3. Numerical results In order to evaluate the performance of the stochastic program, we derive lower and upper bounds from the data given in Hauk et al. (2014). We fix a time horizon of T ¼ 5 since this resembles most of the data, and it allows for fast solvability of the algorithms. As there a only few data points in Hauk et al. (2014) that do not allow to derive meaningful probability distributions for our random variables, we distinguish two settings. First, we assume that all parameters are drawn uniformly at random between the upper and lower bounds given in Table 2 for every time period. Next, we assume that the random variables are sampled from a normal distribution, where the mean is chosen as the center of the intervals given by lower and upper bound, and the standard deviation is chosen such that there is only little mass outside the intervals (the probability that the sampled value is not contained in the interval is below 105). We compare the average profit of the solutions provided by stochastic optimization (SP) with the average profits of: OPT e the optimal offline algorithm knows all increments and requests in advance, and GREEDY e the greedy algorithm acts in every period such that it maximizes the current profit using the available requests. OPT can be determined by solving an integer program as shown in Thielen et al. (2014). Additionally, the algorithm GREEDY was
proposed and turned out to be an algorithm with best-possible competitive ratio for many settings. Observe that we do not consider Algorithm 1 in this section since it is only applicable to a restricted setting (where the online player can sell any amount on the market). As the number of possible combinations of scenarios is too high for a full enumeration, we sample random instances (that consist of one scenario per time period) and use the average profit as an estimate for the expected profit. For a sample size of 10,000 instances we obtain the results presented in the rows with a ¼ 0 in Table 3. Obviously, OPT yields the highest expected profit. The average performance of GREEDY is better than its worst-case performance which only allowed poor competitive ratios (cf. Section 3). However, SP outperforms GREEDY. The better performance of SP follows from the fact that GREEDY does not make any use of the given probability distributions, and SP is tailor-made to guarantee a bestpossible average performance. The advantage of SP becomes even stronger when the probability distributions of the unknown data differ from period to period. This can be seen in the rows for a ¼ 0.2 and a ¼ 1 in Table 3: We modify the scenarios in such a way that all probability distributions remain unchanged, except pmin and pmax are increased in every period by 20% and 100%, respectively, i.e., pmin(t þ 1) ¼ (1 þ a),pmin(t) and pmax(t þ 1) ¼ (1 þ a),pmax(t) for a 2 {0.2,1}. The SP solution tends to save up capacity in the first stages (as it uses the fact that very likely more profitable requests will appear later on), whereas GREEDY simply aims at maximizing the profit in every period. To make our statements more precise, we also performed a onesided t-test for the hypothesis that the objective value of GREEDY exceeds the value of SP. For all settings presented above, we obtained p-values below 105, i.e., the hypothesis can be rejected. The generation of the instances and the dynamic program were implemented in Cþþ using as compiler GCC 4.9.2, and the integer programs for OPT were solved using IBM ILOG CPLEX Optimization Studio 12.5. 5. Conclusion and future work In this work, the economic profitability of short rotation coppice is approached from the point of view of mathematical optimization. In order to incorporate the uncertainties inherent in the determining factors of the problem, i.e., price and biomass yield, the concepts of online optimization and stochastic optimization are applied to the problem. Online algorithms and a stochastic optimization model are proposed, giving possible policies to approach the problem and further insight into the nature of the problem. While previous work on online optimization shows that GREEDY strategies perform well, this is not the end of the story. For a setting with bounded market prices, which has not been researched so far, we present a more sophisticated algorithm THREAT, which we prove to be best possible from the viewpoint of competitive Table 3 Comparison of the average profits (av) and standard deviations (sd) of the solutions obtained by stochastic programming (SP), greedy algorithm (GREEDY), and the optimal offline algorithm (OPT) with price increase factor a 2 {0,0.2,1}.
Table 2 Lower and upper bounds for the evaluation of the stochastic programming solution.
k n p w
min
max
30 1 30 1
60 10 100 100
83
SP
Uniform
Normal
a¼0 a ¼ 0.2 a¼1 a¼0 a ¼ 0.2 a¼1
GREEDY
OPT
av
sd
av
sd
av
sd
625.8 940.6 5096.8 287.5 537.2 4101.9
55.4 51.9 103.1 5.9 7.1 65.1
604.7 863.1 3943.3 281.2 395.2 1764.4
143.9 219.9 1309.6 30.1 57.8 401.8
635.6 984.5 5485.9 293.9 541.0 4123.7
139.2 231.1 1646.8 11.1 22.4 197.4
84
M. Bender et al. / Journal of Cleaner Production 110 (2016) 78e84
analysis. Besides this theoretical result, our simulations indicate that it outperforms GREEDY on randomly generated instances. For the case that we are given probability distributions for the unknown variables, we show how stochastic optimization can be used to determine an optimal policy (with respect to the expected value). Again, our numerical results show that this approach is superior to a simple GREEDY strategy. Both of these findings show that it is worthwhile to design tailor-made algorithms which find optimal (in a certain sense) solutions, rather than simply applying some straightforward heuristic. Future work includes incorporating further aspects of the problem into the mathematical formulation and trying to achieve models which are closer to the real-world problem. For example, it might be interesting to study correlated random variables (in practical application, biomass price and yield are most likely correlated) or discounting effects. While some of the present algorithms are applicable to more general settings, it is not clear how to analyze their performance theoretically or how to design algorithms that take the additional information into account and yield better results. Acknowledgments This research was supported by the German Research Foundation (DFG), grant GRK 1703/1 for the Research Training Group “Resource Efficiency in Interorganizational Networks e Planning Methods to Utilize Renewable Resources”.
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