Three Mathematical Models for Bucking-to-Order

Transcription

1 Silva Fennica 40() research articles ISSN The Finnish Society of Forest Science The Finnish Forest Research Institute Three Mathematical Models for Bucking-to-Order Hamish D. Marshall, Glen Murhy and Kevin Boston Marshall, H.D., Murhy, G. & Boston, K Three mathematical models for bucking-to-order. Silva Fennica 40(): The aim of this aer is to investigate different mathematical aroaches to buck-to-order log merchandizing. A new bucking-to-order lanning model using mixed integer rogramming was develoed to determine the otimal roduction from a stand given different market constraints and forest inventory data. Three different aroaches: market rices, target cutting atterns and adjusted rice list were tested for generating cutting instructions to fulfill the lan created by the new lanning model. The three aroaches were evaluated in four test stands. The market rices aroach simly alied the market rices to each stand. The target cutting atterns aroach alied the samle cutting atterns generated from the lanning model to the stand. The adjusted rice list used a dynamic rogramming algorithm embedded in a search heuristic to adjust both the rices and small end diameters of log roducts to achieve the roduction goals of the lanning models. The results showed that develoing a buck-to-order lan is imortant in obtaining good order fulfillment. The target cutting atterns and adjusted rice list aroaches certainly out erformed the market rices aroach. This aer shows that these two aroaches are caable of achieving excellent order fulfillment. Further develoment and testing is needed to determine which method is the best at generating cutting instructions for buck-to-order merchandizing. Keywords mechanical harvesting/rocessing, otimal bucking, mixed integer rogramming, dynamic rogramming, buck-to-value Author addresses Marshall, Ensis Forests, Private Bag 3020, Rotorua, New Zealand; Murhy, Forest Engineering Deartment, Oregon State University, Corvallis, Oregon 9733, USA; Boston, Forest Engineering Deartment, Oregon State University, Corvallis, Oregon 9733, USA hamish.marshall@ensisjv.com Received 22 December 2004 Revised 9 November 2005 Acceted 23 November 2005 Available at htt:// 27

2 Silva Fennica 40(), 2006 research articles Introduction The adotion of highly mechanized, timber harvesting systems is increasing worldwide (Raymond 988, Nordlund 996, Godin 2000). With these systems, stems are delimbed, bucked and sorted by a single machine. There are a number of factors causing this shift from the traditional motor manual harvesting systems to mechanized harvesting systems. These include economic (the need to continually increase roductivity) and social ressures along with the continuing need to imrove the safety record of forestry oerations. A sometimes overlooked asect of economic imrovement in harvesting is value recovery. Value or revenue can be lost in numerous laces along the forest to mill value chain. One rocess that has been identified as having a large otential for value loss is the rocess of bucking trees into logs. Recent surveys of value recovery studies have shown that on average, manual log making systems were losing % and mechanical log making systems 8% of otential value (Murhy 2003a, Marshall 2005). These kinds of figures have surred significant research in the area of otimal log bucking. Otimal bucking is an effective means of making informed decisions before mistakes are made that result in value loss. A number of mathematical formulations and comuter models have been develoed to otimize the value in each individual stem, this is commonly referred to as buck-tovalue (Pnevmaticos and Mann 972, Briggs 980, Geerts and Twaddle 984, Sessions et al. 989). The objective of buck-to-value otimal bucking is to obtain the maximum monetary value from an individual stem. A stem can be cut u into logs in numerous ways; each set of logs will yield a different financial return. However, there is, in many cases one unique bucking attern that yields the maximum value. The value and logs cut using the otimal bucking attern deends on the secies, diameter, taer rate and quality of the stem lus the roerties and relative market values of log grades being cut. The roblem oerationally is what is otimal for individual stems may not meet the market and oerational constraints at a harvest unit or forest level. To maximize the value coming from a forest, these in-the-field bucking algorithms must be given log secifications which take into account market and oerational constraints. Constraints can be in the form of the following: target volumes, minimum ercentage of volume must be of greater than a certain length, minimum average small end diameter (SED) for a roduct, and minimum ercentage of the volume must be of a certain grade. There may also be constraints on the amount of volume that can be bought from and sold to the oen market. Buying volume from, and selling volume to, the oen market may incur additional costs. In some cases, however, it may be economically better to roduce excess volume of a high value roduct and sell it to the oen market while under-roducing a low value roduct. To account for these oerational and market constraints a number of different buck-to-order rocedures have been develoed. The objective of buck-to-order otimal bucking is to maximize the monetary value at the harvest unit or forest level while meeting market and oerational constraints. Although the majority of buck-to-value models were develoed in the eighties, it has only been in recent years that these models have been imlemented into large scale commercial harvesting oerations (Boston 200). In the literature there are generally two aroaches to develoing the in-the-field cutting instructions for buck-to-order bucking: Aroach : Selecting cutting instructions either before or during the bucking rocess for each tree that will roduce the required volume for each roduct. Aroach 2: Finding the correct rice list (in some cases the correct secifications) that will be alied to the stand of trees to roduce the required volume for each roduct. The first ublished otimal bucking formulation utilized Aroach (Smith and Harrell 96). It solved the buck-to-order roblem using linear rogramming (LP). However, as Pnevmaticos and Mann (972) stated, the Smith and Harrell LP formulation was restricted by the requirement that all relationshis be linear and by the limited number of cutting atterns available for each diameter class. The limited number of cutting atterns issue was solved by Näsberg (985), Mendoza and Bare 28

3 Marshall, Murhy and Boston Three Mathematical Models for Bucking-to-Order (986), Eng et al. (986) and Laroze and Greber (997), by using a two stage iterative formulation of the stand-level buck-to-order roblem. Arce et al. (2002) alied a similar aroach to solve the forest level bucking roblem. The first stage, or master roblem, uses LP to solve the constrained market roblem and the second stage, or subroblem, uses dynamic rogramming (DP) or a network algorithm to solve the individual tree roblem. The shadow rices from the master roblem are used in the second stage to generate new cutting atterns. These are then used to form new columns in the master roblem using column generation techniques. This general aroach is theoretically correct and comutationally efficient (Laroze 993), but as many authors (Sessions et al. 989, Laroze 993 etc) have ointed out, the solutions roduced are not articularly ractical as they roduce large numbers of cutting instructions. Sessions et al. (989) also noted that the requirement of these techniques to subdivide the stand into stem classes makes these solutions hard to imlement. Aroach 2 does not suffer from the same roblems, however, it can not guarantee theoretically that maximum revenue is gained from the bucking of the stand. Duffner (980) is the first ublished work on adjusting the rice list in a bucking algorithm to meet market demands. There is, however, very little detail in the Duffner (980) aer on exactly how he adjusted the rices. Sessions et al. (989) develoed a system to adjust the rices iteratively using a shortest ath algorithm to solve the sub-roblem and a binary search rocedure to find the rice multiliers which obtain the correct ratio of long logs to short logs. The formulation was designed to overcome the roblem of roducing too many short logs that lagued otimal bucking in areas where the Scribner volume scaling rules were used. A number of other rocedures have been tried, such as using an LP solution at the uer level, to adjust the rices in the DP lower level, or using an heuristic to find the correct rices so the demand constraints are met in the master roblem (Laroze and Greber 993, Pickens et al. 992). Laroze and Greber (997) used a rule-based stem bucking algorithm combined with a Tabu Search heuristic to generate easy to imlement bucking rules that are alicable to the entire stand, while roviding the best feasible solution given a set of log rices and market constraints. Laroze and Greber (997) comared the solution from their algorithm with a LP/DP formulation and found that it would lead to rofits aroximately 2.5% below those of the LP/DP algorithm. Laroze (999) used the rule-based aroach described above, in combination with a LP formulation, to solve the forest level bucking otimization roblem. Laroze found that his formulation consistently achieved efficiency levels of aroximately 97% comared to the otimal solutions for all of the scenarios analyzed. Kivinen and Uusitalo (2002) develoed a fuzzy logic controller to adjust the rices secifically for harvesters. The fuzzy logic controller is a set of rules which changes the rice of a log tye based on the disarity between the target roortion and the actual roortion in each log class and the rate of change in this error. Kivinen and Uusitalo found that for the four stands tested, the outut log distribution derived by the fuzzy logic controlled roduction rice matrix was within 92% of the log distribution roduced by the desired (target) rice matrix. Kivinen (2004) ublished another aer outlining a genetic algorithm that searches for rice matrices at the forest level. He found similar results to Kivinen and Uusitalo (2002) Murhy et al. (2004) develoed a two level model where the uer level used a threshold acceting heuristic and the lower level used a DP bucking algorithm. The uer level heuristic was designed to find the roduct rices and minimum SEDs that minimized the difference between the target and the actual log distributions while meeting length roortion and average SED constraints. This algorithm is discussed in more detail later in the aer. The buck-to-order rocess can be slit into three stages; buck-to-order lanning, cutting instruction develoment and adative control during the harvesting. The aim of the research resented in this aer was to develo new, and test existing, algorithms for solving the first two stages of this rocess. The aer resents a new methodology for creating a buck-to-order lan and tests the effectiveness of the two aroaches discussed earlier for creating buck-to-order cutting instructions for imlementation of the buckto-order lan. Three mathematical models have 29

4 Silva Fennica 40(), 2006 research articles been designed to otimise the returns to forest owners, selling into constrained markets. w US i =, K, ( 4) i i z UB i =, K, ( 5) i i 2 Methods The methods section is divided into three arts. The first art describes a new mixed integer rogramming (MIP) formulation for develoing an otimal buck-to-order lan. The second art describes two buck-to-order aroaches, along with a buck-to-value aroach for imlementing the lan. The third art describes the metrics that were used to assess the effectiveness of the aroaches described in the second art. 2. Develoing a Buck-to-Order Planning Model An MIP model was used to maximize the market value of the stand while meeting the market constraints, the customer order book constraints and the sot market constraints. The volume of log roducts that should be cut from the stand was determined. The model otimizes the rojected stand value, given the different market constraints, by determining the otimal bucking atterns for a samle of trees from the stand. The tree data for these samle trees would normally be collected as art of the re-harvest inventory. The model satisfies the customer order book constraints either by using the volume roduced from the stand or buying volume from other sources at an additional cost. In cases where excess volume is roduced from the stand the excess is reallocated to other markets at an additional cost. The mathematical formulation of the model is shown below: Max yi ci + wi di + zi ci zi ei ( ) i= i= i= i= subject to: s xij = yi + wi i =, K, ( 2) j= y + z = b i =, K, ( 3) i i i x ij BigN cut 0 ( 6) ij x MinV cut i =, K, j =, K, s ( 7) ij ij ij x V i =, K, j =, K, s ( 8) ij ij xij CVij i { roduct grou} j =, K, s i cut ij { 0, } ( 9) where = the number of log roducts s = the number of stems b i = the volume demanded of each roduct (i) from the markets x ij = the volume cut of each roduct (i) from each samle stem (j) y i = the volume of each roduct cut from the stand used to fulfill the demand constraints w i = the volume of each roduct sold to other markets. z i = the volume of each roduct bought in from other sources c i = the market rice for log-tye i d i = the sell off rice for log-tye i e i = the buy in rice for log-tye i UB i = uer limit on volume that can be bought from other sources US i = uer limit on volume that can be sold to the markets cut ij = a binary trigger variable, which has a value of 0 if no log-tye i logs are cut from stem j, and otherwise BigN = a large number, for examle 200 Min Vij = the minimum ossible volume for a single log of that log roduct in that stem (It is found by otimal bucking the stem using only that roduct and restricting the length of the logs to the smallest ossible length for that log roduct) V ij = this is the maximum otential volume that can be cut from stem (j) of that roduct (i) (This value is found by bucking the stem using a dynamic rogramming bucking algorithm, using only the roduct secifications for that 30

5 Marshall, Murhy and Boston Three Mathematical Models for Bucking-to-Order Table. An examle of downgrade grous. Product Minimum small end Allowable Member of roduct downgrade grous diameter SED (mm) qualities Pruned 350 A Pruned Sawlog 200 AB Pruned, Sawlog Sawlog ABC Pruned, Sawlog 2 Sawlog ABC Pruned, Sawlog, Sawlog 2, Sawlog 3 Pul 50 ABCD Pruned, Sawlog, Sawlog 2, Sawlog 3, Pul Waste 0 ABCDE Pruned, Sawlog, Sawlog 2, Sawlog 3, Pul Waste roduct and waste, where waste has a value of zero in this model) CV ij = the maximum constrained volume; this is the maximum volume from a stem when all the roducts in a articular roduct s downgrade grou are used in the bucking algorithm. A downgrade grou is defined as those roducts that can be downgraded based on quality and small end diameter secification into that roduct. Examles of roduct grous and their definitions are given in Table. The constraints shown above ensure that: Eq. 2 The sum of the volumes cut from all the stems for each log roduct, is equal to the volume roduced from the stand that is being used towards fulfilling the order, lus the volume being sold onto the oen market. Eq. 3 The sum of the volume roduced from the stand that is being used towards fulfilling the order, lus the buy in volume, is equal to the demand requirement for each log roduct. Eq. 4 The amount of volume that can be sold on the oen market is limited. Eq.5 The amount of volume that can be bought from the oen market is limited. Eq. 6 A binary trigger is set to either 0 or. If x ij is greater than zero then cut ij must be. Combining this constraint with the constraint in Eq. 7, requires x ij to be greater than Min Vi. Eq. 7 The volume of log roduct (i) cut from a stem is greater than integer multiles of the minimum log roduct volume. Eq. 8 The total volume of all logs of log roduct (i) cut from stem (j) is less than or equal to the maximum otential volume for that log roduct in that stem. Eq. 9 The total volume of the downgrade grou is less than or equal to the maximum otential volume for that downgrade grou in that stem. The MIP model was formulated in AMPL mathematical rogramming language and solved using CPLEX 8.0. The default CPLEX otimizing settings were used. The model was solved on a Pentium 4 lato with GB of memory. Most of the models took less than one minute to solve. The solution rovides rojected volumes that should be ) cut from the stand, 2) be sold on to the oen market, and 3) urchased from the oen market to satisfy the order book constraints. Solving the MIP formulation also creates cutting atterns for each of the samle trees included in the formulation. These cutting atterns, which are in the form of volume er roduct (x ij ) and the maximum otential volumes er roduct (V ij ) for each tree, are used in the Cutting Pattern aroach described in the next section. 2.2 Methods for Imlementing the Buck-to- Order Plan 2.2. Aroach : Market Price In this aroach the market rices were alied using an individual stem otimal bucking DP algorithm. The basic formulation has been ublished by a number of authors in the ast (Pnevmaticos and Mann 972, Briggs 980, Geerts and Twaddle 984 and others). This algorithm was develo by the first author of this aer and is similar to that described by Deadman and Goulding (979). Its aim was to maximize the total value of each stem by determining the otimal allocation of the log 3

6 Silva Fennica 40(), 2006 research articles roducts on a stem. The log roducts are secified in terms of maximum and minimum SED (small end diameter) and LED (large end diameter), feasible lengths and minimum quality requirements. The quality requirements were secified as a single character code which reresents a combination of branch size and swee characteristics. The algorithm was alied to all the trees in the stands using the roduct market rices given in Table 3. This is effectively a buck-to-value aroach and will otimize the market value for each stem in the forest. The roduct volume for the stand is determined by adding u the volume for each roduct for all the trees in the stand Aroach 2: Target Cutting Patterns This algorithm was formulated to solve the roblem of determining which bucking attern should be alied to each tree, by allocating the target cutting atterns to a tree using the maximum otential volume (V ij ) of each log roduct. The theory is that two trees that are similar in terms of size and quality characteristics will have similar maximum otential volumes and hence should be bucked in a similar manner. The theory behind this formulation is that each of the target trees in the inventory samle reresents the same roortion of trees in the total stand and the target cutting attern will therefore, be alied to the same roortion of trees in the stand. The target cutting attern to be used on the current stem is found by determining which of the samle trees is most closely matched in terms of maximum otential volumes. A simle distance function, as shown in Eq. 0, was used to determine the nearest neighbours: n d = PTV i PAVi i= where d = the distance between trees in terms of total otential volume for each roduct i = roducts (,,n) PTV i = the maximum otential target volume for roduct i PAV i = the maximum otential actual volume for roduct i for the current stem. ( 0) Ponsse harvesters store information on the revious 80 stems harvested and use the nearest eight trees as the basis for redicting stem taer rather than the taer from the single closest tree. It is ossible to use k-nearest neighbours to determine the best cutting attern to aly to each tree in the stand. The same distance function (Eq. 0) was used to determine the k-nearest neighbours for the current candidate tree. Trials using different numbers of nearest neighbours (k) showed that no gains were made by using more than 4 of the closest trees (in terms of d) for this alication. The target volumes for each roduct are then calculated from the k-nearest neighbours. The target volumes are calculated in roortion to the distance each of the k-nearest neighbours is from the current stem. The following equation is used to calculate the target volumes for the current stem: k k TVi = TV im dm / d m / ( k ) m= m= () where i = roducts (,,n) m = nearest neighbours (,,k) d = the distance between trees in terms of total otential volume (as calculated in Eq. 0) TV i = the target volumes for the current stem TV im = the target volumes from the k-nearest neighbours. The target volumes for each roduct are then used in a heuristic allocation model that uses the same structure as a forward recursive DP algorithm to minimize the deviation from these target volumes. The decision whether to cut a roduct at a articular state in this roblem deends on what roducts have been cut before, hence breaking the rincile of otimality. The algorithm attemts, as closely as ossible, to cut the same volumes out of the current tree as the samle cutting attern. This is achieved by relacing the maximize revenue objective function with a minimize the weighted volume deviation from the target volume objective function. The minimize volume deviation requires the addition of i more state variables; where i reresents the number of roducts. These new state variables contain the volume of each roduct that has already been cut at the state. 32

7 Marshall, Murhy and Boston Three Mathematical Models for Bucking-to-Order The volume deviations from the target of each roduct are weighted to encourage the algorithm to meet the targets for higher value roducts. This is done using the market rices for the roducts. If the current cut volume of that roduct at that stage is less than the target volume for that roduct, the market rice for that roduct is used as the weight. However, if the current cut volume of that roduct at that stage exceeds the target volume then the market rice list is alied in reverse. During the early testing of this algorithm it was found that the ul and waste roducts were being over roduced. In an attemt to reduce this, the target volumes for all the target cutting atterns for ul and waste were set to zero. This changed the objective function of the algorithm to minimize the volume deviation from the target volumes (for all roducts excet ul and waste) while minimizing the roduction of ul and waste volumes. This change significantly imroved the erformance of the algorithm Aroach 3: Adjusted Price List The adjusted rice list algorithm that has been used in this aer is FASTBUCK. This was develoed by Murhy et al. (2004). In this algorithm an individual stem otimal bucking DP rocedure is imbedded within a threshold acceting algorithm which adjusts relative rices for log roducts to meet order book constraints. The threshold acceting algorithm is designed to otimize the order fulfillment, not the market value of the volume roduced. The objective function is to maximize the aortionment degree, which is a measure of how well the roduction meets the orders. Aortionment degree (AD%) is defined as: m Ddi D AD% = 00 i= 2 i ( 2) where AD% = aortionment degree (goodness of fit between the demand and roduction vector/ matrix) m D di D i = number of log grades = target roortion demanded for the log grade = actual roortion roduced for the log grade. A set of good relative rices is found through an iterative rocess of changing the relative rices. The DP bucking algorithm bucks each of the stems in the samle given a set of relative rices. The AD% is calculated for the resulting volumes generated from bucking the set of samle stems. If the AD% is better than, or within a certain threshold of, the current best AD%, the current set of relative rices is ket as the starting oint for the next iteration. If the AD% is outside the threshold, the current set of relative rices is discarded. Only one roduct s rice is changed at any one time. The roduct is randomly selected and its rice changed by increments of $. The rocess stos when a set number of iterations have been comleted (Murhy et al. 2004). In this aer the target roortion of the total volume for each roduct was determined using the results from the buck-to-order lanning model. The rojected volume roduction for each roduct was divided by the total rojected volume. The FASTBUCK algorithm was first alied to the reharvest inventory stem data. The resulting relative rices and minimum SED were then alied to the whole stand to simulate the harvesting rocess. 2.3 Calculating the Effectiveness of the Aroaches The roduction from the simulated harvest from each aroach was then adjusted using the rojected buy in and sell off volume from the Buck-to-Order lanning model. The effectiveness of the different aroaches were measured using ) the level to which the orders were fulfilled and 2) the monetary return from harvesting the block. The metrics that were used are described below: 2.3. Order Fulfillment To evaluate the goodness of fit between the demand and roduction vector/matrix the aointment degree (AD% in Eq. 2) was used. It was originally develoed by Bergstrand (989). This 33

8 Silva Fennica 40(), 2006 research articles is a commonly used measure for evaluating the fit between an actual outut distribution of logs and the desired log distribution Monetary Return The monetary return is calculated by determining the gross value gained by harvesting the unit, given that the volume of the over roduced roducts is sold on to the oen market at a discounted rice (Table 4), and the orders that are undersulied have to be fulfilled using volume bought from the oen market at a inflated rice. The monetary return (MR) is determined using the following equation: MR = Max y c + w d + z c z e ( 3) i i i i i i i= i= i= i= where the coefficients are the same as those defined for Eq.. The formula gives the monetary return of the solution, given that the log demand distribution has been comletely fulfilled, either from the stand, or from the lanned sales and urchases from the oen market. However, given that erfect information is not available, it is ossible that additional volume will have to be urchased from the oen market during or after the harvest. These urchases come at a significant cost to the comany. In this aer, it has been assumed this cost will be 25% of the original market rices. Any volume roduced in excess of the originally rojected roduction is valued at the rice of ul, regardless of its original value. 3 Materials 3. Test Stands Four stands were used to test and evaluate the erformance of the above buck-to-order lanning and imlementation aroaches. The four stands were the same as those used in Murhy et al. (2004). All stands had been runed and were of similar mean diameter at breast height (DBH), details of which are rovided in Table 2. Only one of the i i Table 2. Characteristics of test stands. EVEN UNEVEN FROST WHAKA Total area (ha) Density (stems er ha) Mean DBH (cm) Total volume (m 3 ) four stands was a real-world stand; this stand (WHAKA) was a Pinus radiata lantation stand in the North Island of New Zealand. Every tree in this irregular shaed stand was located, measured and described using the MARVL inventory system (Deadman and Goulding 979). The other three stands were virtual stands and were rectangular in shae (500 m 200 m). These were based on growth and form characteristics of Pinus radiata and were generated to reresent a variety of forest conditions. The lower limbs had been removed (runed) from all trees to a height of aroximately 6 m in the EVEN stand. Selection for runing was uneven in the UNEVEN stand; 00% of trees were runed in the middle of the stand decreasing to 70% at the edges of the stand. This mimicked situations where the runing contract suervision or funds were inadequate to ensure all final cro trees in the stand were runed. The EVEN and UNEVEN stands were generated to have diameter distributions which ranged from 20 to 70 cm with a DBH of aroximately 45 cm and standard deviation of 5.9 cm. The FROST stand mimicked a situation where there was a frost effect in the center of the stand; tree size was small in the center and increased toward the edge of the stand. All trees were runed in the FROST stand. Fifteen circular re-harvest inventory lots were systematically located in each of the EVEN, UNEVEN, and FROST stands and five square lots were located in the WHAKA stand. The inventory lots occuied 3% of total area in each stand. 3.2 Product Requirements for Test Stands The same roduct requirements were alied to all four test stands (Table 3). There were five log-tyes (Pruned Domestic Sawlogs, Unruned 34

9 Marshall, Murhy and Boston Three Mathematical Models for Bucking-to-Order Table 3. Market requirements and constraints for the four test stands. Log-tyes Lengths Minimum Market Sell off Buy in Target SED rices rices rices roortions (m) (mm) ($/m 3 ) ($/m 3 ) ($/m 3 ) (%) Pruned Domestic Sawlog Unruned Exort Sawlog 2.2 m Unruned Exort Sawlog 8.2 m Domestic Sawlog # Domestic Sawlog #2 Longs Domestic Sawlog #2 Shorts Pul Waste Exort Sawlogs, Domestic Sawlogs #, Domestic Sawlogs #2, and Pul) lus waste. Most log tyes allowed multile lengths; some in multiles of 0.3 m, others in multiles of 0. m. A total of 5 lengths were included in the analyses. Each logtye had target roortions of total volume that were required. For examle, the demand target roortion for Pruned Domestic Sawlogs was 5%, of the total volume harvested. Three different rices were included in the secifications for each log tye: The market rices, which were the rices for the volume of each log tye with confirmed markets. The sell off rices that can be thought of as, either a transfer cost into log stocks, or the rice received for selling excess volume on the oen market. These rices were 5% less than the market rices rounded to the nearest dollar. The buy in rices can also be thought of as a transfer cost out of log stocks, or the rice for buying volume from the oen market. These rices were 0% greater than the market rices rounded to the nearest dollar. Waste was given a negative sell off rice of $ and ositive buy in cost of $ to reresent the cost of handling under and over roduction of waste volume. 3.3 Market Scenarios To test the robustness of the different aroaches four different test market conditions were used: ) Unconstrained Sot Markets (Unconstrained Sot) This is the base scenario; it uses the rices in Table 4. The scenario has no constraints on the volume that can be brought in from, and sold off to, the oen (also referred to as sot ) market. 2) High Buy In for Unruned Exort Sawlog 2.2 m (Hi-Price Ex 2) In this scenario the buy in rice for the Unruned Exort Sawlog 2.2 m was increased from $07 to $47 which is greater than the Sell Off rice of Pruned Domestic Saw. This is to simulate an increase in rice on the sot market due to limitations in the suly of Unruned Exort Sawlog 2.2 m. 3) Sot Market Availability Constraints (Sot Constraints) The volumes for EVEN, UNEVEN and FROST stands of Unruned Exort Sawlog 2.2 m and Domestic Sawlog #2 Shorts that was available on the sot market is limited to 575 m 3 and 300 m 3 resectively. The available market for the surlus Pruned Domestic Saw volume was limited to 75 m 3. The numbers were reduced to 30, 3 and 70 for the WHAKA stand. These volumes were chosen arbitrarily, solely to constrain the model. 4) Minimize Buy In and Selling Off volume. (Buy/Sell Min) To test the robustness of the model, the objective function of the buck-to-order lanning model was changed from maximizing return to minimizing the amount of the volume that was brought in and sold off. To sto that model just roducing lots of Waste, the objective function was formulated to 35

10 Silva Fennica 40(), 2006 research articles minimize the roduction of waste as well. Min wi + zi + y7 ( 4) i= i= where y 7 = volume of waste. 4 Results 4. Buck-to-Order Planning Model The buck-to-order MIP lanning model was run on all four stands under the four market scenarios. Fig. shows the rojected maximum values for each stand and market scenario combination. The constraint on buy in volume of Unruned Exort Sawlog 2.2 m had to be relaxed for the UNEVEN stand under the Sot Constraints scenario, as the model was infeasible with the original constraints. The chance of obtaining an infeasible solution will be significantly increased as the number of hard constraints that are laced on the amount of available buy in and sell off volume increases. The market value objective function shows the comany s oerational lanner and marketers the trade offs from lacing more market constraints on a stand. For examle, trying to minimize the buy in and sell off volumes reduced the market value of the forest for the UNEVEN stand by 5% comared with the unconstrained sot market value. The effect on the volumes that are cut from the EVEN stand, under the four market scenarios is shown in Fig. 2. The grah shows the change in the roortion of the total stand volume that is rojected to be cut for each roduct from the stand in comarison to the original order book targets given in Table 4. Increasing the buy in cost of the Unruned Exort Sawlog 2.2 m caused the model to cut more of that volume from the stand, reducing the volume of Pruned Domestic Saw log, and the overall return from the stand by 7%. Minimizing the amount of buy in and sell off volume was the most costly scenario of the four market scenarios for all four stands. 4.2 Imlementing the Buck-to-Order Plan 4.2. Order Fulfillment Effectiveness The AD% from simulating imlementation of the different aroaches to carrying out the lans for each market scenario is given in Table 4. All three Fig.. The rojected objective function ($/ha) from the buck-to-order lanning model for each stand under the different market scenarios. 36

11 Marshall, Murhy and Boston Three Mathematical Models for Bucking-to-Order Fig. 2. The rojected roortion of the different roducts being cut from the EVEN stand, under the different market scenarios. Table 4. Aortionment degree for three aroaches for imlementing a buck-to-order lan. Stand Scenario Aroach Market Price Cutting Pattern Adjusted Price List EVEN Unconstrained Sot 92.5% 93.6% 95.4% Hi-Price Ex_2 88.0% 92.% 95.% Sot Constraints 87.8% 9.6% 94.0% Buy/Sell Min 86.2% 94.% 96.4% UNEVEN Unconstrained Sot 94.6% 92.4% 92.4% Hi-Price Ex_2 90.0% 9.3% 95.2% Sot Constraints 90.0% 9.3% 96.0% Buy/Sell Min 89.5% 94.2% 95.2% FROST Unconstrained Sot 9.0% 88.% 90.5% Hi-Price Ex_2 82.4% 86.5% 88.0% Sot Constraints 82.4% 84.9% 89.4% Buy/Sell Min 84.8% 9.% 86.6% WHAKA Unconstrained Sot 59.0% 74.8% 80.5% Hi-Price Ex_2 57.6% 67.6% 75.2% Sot Constraints 52.0% 52.3% 77.8% Buy/Sell Min 65.3% 73.0% 8.5% aroaches did well in the unconstrained sot market scenario. The order book constraints can simly be fulfilled by buying in needed volume and selling off excess volume to the sot market. As additional market constraints were laced on how the order could be fulfilled, erformance of the Market Price aroach decreased while the other two aroaches continued to roduce high AD%. In the Market Aroach there is no way to adat the cutting instructions to take account of the additional constraints. Overall the three virtual stands roduced sub- 37

12 Silva Fennica 40(), 2006 research articles Table 5. The best aroach in terms of aortionment degree for each market scenario and stand combination. Unconstrained Sot Hi-Price Ex_2 Sot Constraints Buy/Sell Min EVEN Adjusted Price List Market Price Adjusted Price List Adjusted Price List UNEVEN Market Price Adjusted Price List Adjusted Price List Adjusted Price List FROST Adjusted Price List Adjusted Price List Adjusted Price List Cutting Pattern WHAKA Adjusted Price List Adjusted Price List Adjusted Price List Adjusted Price List Table 6. The best aroach in terms of monetary return for each market scenario and stand combination. Unconstrained Sot Hi-Price Ex_2 Sot Constraints Buy/Sell Min EVEN Cutting Pattern Market Price Cutting Pattern Cutting Pattern UNEVEN Market Price Adjusted Price List Adjusted Price List Adjusted Price List FROST Market Price Cutting Pattern Cutting Pattern Cutting Pattern WHAKA Adjusted Price List Cutting Pattern Adjusted Price List Cutting Pattern stantially better AD% than the small Whaka stand. This difference in AD% is robably as much a function of the small inventory samle size as a function of the algorithms. The effect of the increased within-stand variation can be seen in all three aroaches. The AD% is much lower in the FROST stand than the EVEN stand. The Adjusted Price List aroach generally out erforms the other two aroaches (Table 5). It roduced substantially better results in the WHAKA stand, although this may again be as much a function of the small inventory samle size as a function of the algorithms Monetary Return When monetary return is used as the metric for comaring imlementation aroaches, the Adjusted Price List aroach did not consistently outerform the others. Having the highest AD% does not guarantee the highest monetary return from the stand. This is because the AD% metric treats every log roduct equally, however, rarely is the imortance of fulfilling every order the same. Table 6 summarizes the monetary return results, showing which aroach achieved the highest monetary return for each stand under the different scenarios. Using this metric to evaluate the erformance of the different aroaches, the Cutting Pattern aroach generally outerformed the other aroaches but may be stand and market scenario deendent. The advantage, in terms of monetary return, of develoing a buck-to-order lan is shown in Fig. 3. It shows, for the Buy/Sell Min market scenario, the ercentage imrovement in return of the three different aroaches when a buck-to-order lan has been develoed comared with when no lan has been develoed. Only the Adjusted Price List aroach in the WHAKA stand does not roduce a ositive ercentage increase in revenue. This is because, in order to maximize the AD%, the model under-roduced the high value roducts and over-roduced the low value roducts. In this case the two objectives, monetary return and order fulfillment, were in conflict with each other. 5 Discussion and Conclusions An otimal bucking olicy can be roduced for a single stem, a single stand, or a set of stands to be successively or concurrently harvested (a forest) (Kivinen and Uusitalo 2002). We have introduced three single-stand models that could be used in the lanning for, and imlementation of, bucking-to-order rocedures on mechanised harvesters. 38

13 Marshall, Murhy and Boston Three Mathematical Models for Bucking-to-Order Fig. 3. The ercentage increase in return using the different aroaches under the Buy/Sell Min marketing scenario. The models were only alied to a single secies, Pinus radiata which, unlike Norway sruce (Picea abies), often exhibits considerable variation in quality along each stem. The models were tested in four stands of similar average size but differing levels of comlexity in terms of stand treatment (thinning and runing) and tree size distribution. Only one of these stands was a real-world stand and it was rather small, less than 2 ha. A single set of seven log-tyes and a single target distribution were alied to all four stands. In comarison with Scandinavian markets, where there are many log classes based on combinations of length and diameter, demand constraints were relatively few. Performance of the models was tested under four levels of market restriction, however. In ractice, modern harvesters are able to not only measure length and diameter but also redict the rofile of the unknown art of the stem. In a normal oeration, the harvester head measures and delimbs a ortion of the stem length, then the harvester s comuter redicts the rofile for the rest of the stem. An otimal solution is then calculated and a log is cut. The rocess is then reeated until the whole stem has been comletely cut into logs. In our models we assume that ) all stem measurements are error-free and 2) the entire stem, not just a ortion of it, is measured rior to calculating an otimal bucking solution. Marshall (2005) and Murhy (2003b) have shown that both these assumtions are likely to result in log distributions that will differ from those found in ractice. Rather than have to decide on what would be aroriate error distributions and scanning lengths to use we decided to control these source of variability for this study. Given the above limitations of this study we have been able to show that ) significant gains can be made by first determining the otimal volume that should be cut from the stand; that is, buck-to-order lanning, and 2) the Target Cutting Pattern and Adjusted Price List aroaches will generally outerform the simle Market Prices aroach for imlementing the buck-to-order lan. The buck-to-order lanning model gives harvest lanners the ability to analyze different market and oerational conditions before harvesting the stand. It not only maximizes the value of the stand, given that market constraints exist, but also rovides redictions of the surlus volume and the required extra volume before starting to harvest the stand. Having this information means that good markets can be found for the surlus volume 39

14 Silva Fennica 40(), 2006 research articles as well as otential sources for the volume that is going to be in short suly. Having to buy volume off the sot market can sometimes be extremely costly. Equally as costly is having surlus volume that can not be sold. Often, unsold volume has to be downgraded to a lower value roduct that can be sold. The buck-to-order lanning model also enables the lanner to determine the costs of forcing the stand to roduce sub-otimal target distributions. The buck-to-order lan formulation resented in this aer is for a single harvesting/rocessing machine oerating in a single stand. There are a number of ossible extensions to the formulation that would better model the oortunities that may be available to real-world lanners. Kivinen (2004) reorts that simulations carried out by Imonen (999) have shown that harvesters linked through mobile communications and sharing data can more raidly achieve a high AD% than the same harvesters without this real-time connection. Kivinen (2004) also indicated the oututs from multile stands may comlement each other and facilitate meeting market constraints. Since not all logging crews have the same roduction caabilities, there is the otential to also integrate the crew scheduling concets resented in Murhy (998) and Mitchell (2004). These extensions to the buck-to-order roblem, that is taking into consideration multile machines, multile stands and crew scheduling, could be added to our formulation. They would, however, increase the roblem size dramatically, and may require the use of column generation techniques to solve the lanning roblem. Imlementing the buck-to-order lan is not easy. We have shown that using market rices will generally not result in the best imlementation of the lan. In some comanies, harvest schedulers adjust the market rices without the use of comuter aids to take into account market constraints. Although the basic idea behind the adjustment rocess is quite obvious, Kivinen and Uusitalo (2002) reort that there have been marked differences in bucking results between regions, contractors, and harvester tyes in Finland from using such an aroach. We found that the Adjusted Price List model seems to out erform the Cutting Pattern model when AD% is used as a metric. Although not entirely unambiguous, it seems that this is reversed when the monetary return is used as the metric of erformance. This is largely due to the objective functions in the two algorithms. The Cutting Pattern model weights the volume deviation to lace a high imortance on cutting the higher value roducts, whereas the Adjusted Price List model simly tries to maximize the AD%. AD%s for the three virtual Pinus radiata stands ranged between 84.9 and 96.4% and deended on both market scenarios and stand conditions. These are similar to those reorted by Malinen and Palander (2004) for seven virtual, sruce-dominated stands and by Kivinen and Uusitalo (2002) for four real-world, mature Norway sruce stands. Malinen and Palander used DP and a near-otimum aroach for selecting cutting atterns for each tree. Kivinen and Uusitalo (2002) used DP and fuzzy logic to adat the rice list used for cutting trees within each stand. The AD% s for the real-world WHAKA stand ranged between 52.3 and 8.5%. We believe that the low AD%s for this stand are robably, more a function of the small samle size of the inventory (4 trees in total), than the cutting instructions and targets generated by the buck-to-order lan. Further work is required to determine the otimal samle size for generating the buck-to-order lan and the cutting instructions for fulfilling the lan. The buck-to-order roblem is relatively easy to solve with erfect information on all the trees in the stand, however obtaining this information can be extremely costly. Further imrovements to both the Adative Price List and Cutting Pattern models are undoubtedly ossible. Malinen and Palander (2004) have demonstrated, for examle, that using a flexible, enalty-segmented AD% metric can lead to overall imrovements in the AD%. It would also allow weighting of higher value roducts and may lead to higher monetary return values although we have not tested this. We were able to show, however, that significantly better order fulfillment was achieved for all four market scenarios by the Cutting Pattern model when ul and waste volume targets were set very close to zero. It is difficult to know, without further testing, whether this is a universal rule or just alies to the stands/market scenarios used in this aer. Many harvesters on the market have adative 40

15 Marshall, Murhy and Boston Three Mathematical Models for Bucking-to-Order buck-to-order systems installed. These systems adjust the cutting instruction while working through the stand. For examle, the Ponsse s comuter uses an adative-rice list where the value of each log grade is changed, as the harvesting rogresses through the stand, in accordance with how well the demand for each roduct is being met (Sondell et al. 2002). Other harvester comuter manufacturers, such as Dasa4, Timbermatic 300, Valmet and Motomit, have imlemented an aroach called close-to-otimal, where a cutting solution is selected from the to 5 % of the buck-to-value solutions that best fulfills the demand requirements (Sondell et al. 2002, Kivinen and Uusitalo 2002). Sondell et al. (2002) reort AD%s of slightly over 80% for these systems. Our three models used re-harvest inventory data for each stand to develo the buck-to-order lan, targets, adative rice, and target cutting atterns. Other stem data sources could also be used. Kivinen and Uusitalo (2002) found that using inventory data in their fuzzy logic model to find the adative rice list gave the highest AD% for only one of the four sruce stands tested. In the other three stands, using only data from revious stems harvested to adjust the rices roduced the best AD%. Similarly, Murhy et al. (2004) found that using re-harvest inventory data from four Pinus radiata stands resulted in AD%s that were 0.7 to 7.6 oints lower than when using recently harvested stem data. Order book constraints that constrain the total volume of the different log roducts are not the only tye of market constraints. Other log mix constraints, such as minimum average SED and ercentage long logs within a roduct grou are required by some customers. These tyes of constraints were not included in the analysis carried out in this aer. However, the FASTBUCK algorithm (Murhy et al., 2004), was develoed so that these tyes of constraints could be included. It is feasible, yet not tested, that these constraints could be included into the formulation of Cutting Pattern model. The minimize volume deviation objective function in the dynamic rogramming algorithm could be enalized if a articular log caused the constraints to be violated. Three models for bucking-to-order were described in this aer; one for buck-to-order lanning and two for buck-to-order imlementation. The buck-to-order lanning and imlementation models were shown, in four test stands and a range of market scenarios, to yield log distributions that more closely met target distributions than would be found when using market rices alone. Further testing of these models in a wider range of stands, secies and market conditions is required. Recent work in Scandinavia, USA and New Zealand indicates that further develoment of models such as these is both ossible and likely to further imrove their utility. References Arce, J.E., Carnieri, C., Sanquetta, C.R. & Filho, A.F A forest level bucking otimization system that considers customer s demand and transortation costs. Forest Science 48(3): Bergstrand, K.G Fördelningsatering med närotimal. Skogsarbeten. Unublished internal reort.. Boston, K Precision log making for lantation oerations. Proceedings of the First International Precision Forestry Cooerative Symosium, June 200 Seattle, Washington, USA Briggs, D.G A dynamic rogramming aroach to otimizing stem conversion. Ph.D. thesis. University of Washington Deadman, M.W. & Goulding, C.J A method for assessment of recoverable volume by log-tye. New Zealand Journal of Forestry Science 9: 6 75 Duffner, W.W Decision making from market to stum. Proceedings of Weyerhaeuser Science Symosium, Tacoma, Washington, USA Eng, G., Daellenbach, H. & Whyte, A.G.D Bucking tree-length stems otimally. Canadian Journal of Forest Research 6: Geerts, J.M.P. & Twaddle, A.A A method to assess log value loss caused by cross-cutting ractice on the skidsite. New Zealand Journal of Forestry 29(2): Godin, A.E Logging equiment database: 999 udate. Forest Engineering Research Institute of Canada, Advantage (20). 2. Imonen, V Puutavaralogistiikka elkistää hankinnan toimintamallit. Metsätieteen aikakauskirja 4: (In Finnish). 4