Improved supply chain management using spatial statistics
J.D. Hamann1 and K. Boston2
1
Department of Forest Engineering, College of Forestry, Oregon State University,
Corvallis, Oregon, USA 97330 e-mail: jeffery.hamann@oregonstate.edu
2
Department of Forest Engineering, College of Forestry, Oregon State University,
Corvallis, Oregon, USA 97330 e-mail: kevin.boston@oregonstate.edu
_______________________________________________________________________
Abstract
The benefits of spatial inventories are numerous and allow for the direct
development of precision harvest planning; thus, maximizing customer service
by better identifying areas within harvest units that can be processed to meet
customer orders while minimizing handling, processing, and storage costs.
Simulations were performed to examine potential benefits of using Kriging to
estimate product volumes in unsampled areas of a harvest block. Using a stem
map, we sampled the area using 0.01 ha fixed radius plots on a 22.2 × 16.6
meter grid to obtain sawlog volume estimates with and without spatial
information.
To demonstrate an advantage of a spatially explicit sampling method, we then
optimized the harvesting operations, using a simulated annealing algorithm, for
a fictitious operation so that the optimal harvest pattern that would minimize the
sum of the squared deviations, between the demand and the predicted
production from the harvest block.
_______________________________________________________________________
Introduction
Log manufacturing is a disaggregative manufacturing process (e.g. meat production,
agriculture, mineral extraction) which is almost always is associated with many types of
variation. The quality of and quantity of the raw material within a given order may vary within
certain predefined levels such as diameter ranges, lengths or surface characteristics and
unlike finished panel products or boards, which have a minimal variation, the production of
the raw material is subject to variation within the stem, stand and season. To reduce this
variation, forest products companies have sought to optimize the sample size for a desired
level of precision (Oderwald and Jones, 1992; Brooks and Wiant, 2004; Zeide, 1980; Gambill
et al., 1985) , increase the level of detail in sampling procedures by measuring additional
attributes (Mandallez and Ye, 1999) and by including additional data sources in the inventory
(Holmström, 2002; Kilkki and Päivinen, 1987; Korhonen and Kangas, 1997). While obtaining
more detailed information on operational harvest units can be costly, it can lead to significant
improvements in the financial performance of an integrated forest products firm as log mix
can be optimally matched with processing facilities (Wagner et al., 1996; Uusitalo, 1997).
Variance reduction, it then seems, is the most important task for primary forest supply chain
management and optimization and since the spatial component of forests account for the
most variation regarding the production of logs, it then seems appropriate to examine the
spatial aspect of log production (Korhonen and Kangas, 1997; Uusitalo, 1997; Rasinmäki
and Melkas, 2005). Until recently, there has been little work that examines the addition of
spatial data in sampling to reduce the variation in production estimates. Murphy et al. (2004)
found that they could achieve a 17 to 22 percent increase in stand value, from a 3 percent
sample of a radiata pine (Pinus radiata D. Don) stand in New Zealand. The authors
concluded that harvest manager’s ability to capitalize on within stand variation (harvest
pattern selection) and the inability control order book requirements (constraints), it might be
possible to reduce variation in production estimates by simply modifying the harvest pattern.
Choosing an operational pattern that reduces variation in the delivery of raw materials is not
new. Mining geology (Journel and Huijbregts, 1978), and to a lesser extent soil science
(Kravchenko, 2003), have used spatial prediction and mapping for short term planning
where, unlike commonly practiced sampling methods in forestry, variance estimates are
generated for unsampled areas. While mining geology has accepted the high cost of
inventory and data collection, forest managers have emphasized cost reduction using a
variety of methods such as reducing sampling intensity, utilization of remote sensing for
imputation or increasing remeasurement periods. In contrast, Scandinavian forest managers
have began to examine the additional benefits of high resolution data from spatial prediction
methods (Holmström, 2002; Korhonen and Kangas, 1997) or from stem map data obtained
during processing in harvester operations (Stendahl and Dahlin, 2002; Rasinmäki and
Melkas, 2005).
This paper examines how spatial information can be added to traditional sampling
procedures to increase the usefulness of the sample while maintaining or possibly increasing
precision in the context of the primary forestry supply chain. A simple scheduling model is
presented that demonstrates the additional gains from the spatial analysis and briefly
discuss the advantages and disadvantages of the additional resources and processing
required to obtain the spatially explicit log volume information within a harvest area.
Data and Methods
Study Site and Stem Map
The stem map used for this analysis was obtained from the NASA Forest Ecosystem
Dynamics (FED) project (Walthall et al., 1993). The study site is a three hectare rectangle
(150m × 200m) located 56 km north of Bangor, Maine in Penobscot County (45 12’N, 68
44’W). The original data file contained 5967 tree records. Each tree record contained x and y
coordinates, species, diameter at breast height (DBH), total height (THT), canopy position
and an indicator variable to represent a dead stem. We selected only those trees that were
living, over 8 cm and standing for our study and for simplification assumed all trees were a
single species. The final number of stems within the 3 hectare stem map was 4390. The
stems were then bucked into log lengths of four meters, to a 2 cm top, using the taper
equation presented by Kozak et al. (1968) with a stump height of 0.3 m and 0.2 m of trim.
The volume of all logs with a small end diameter over 10 cm was totaled as the sawlog
volume
Sampling Methods
To determine the base case for comparing the non-spatial (traditional) and and spatial
sampling methods, we simulated a commonly applied square grid sample design of 0.01 ha
fixed-radius plots on a 22.2 × 16.6 meter grid. The distance of each stem to the plot center
was computed to determine total sawlog volume per plot.
To determine the benefits of adding spatial information, we used the plot locations from the
grid generated during the non-spatial sampling phase with the plot summaries to then predict
the sawlog volume in unsampled locations. After visual examination of the stem map, we
used an exponential variogram with universal Kriging to predict the sawlog volume over the
entire area. Model fitting and prediction were performed using the gstat package within R
(Ihaka and Gentleman, 1996; Pebesma, 2004). We used the methods described by Journel
and Huijbregts (1978) and Kim and Baafi (1984) to combine the variance estimates from all
the individual predicted cells.
Demand Requirements and Production
To evaluate the estimates from Kriging, we compared the deviations between a generated
random demand function for a 12 day production period and the predicted sawlog volumes
from various harvest patterns. To make the demand levels as realistic as possible, the
average production required was the sum of the total volume for the area divided by 12
production days. Production requirements were varied from a minimum of 50 m3 to 140 m3
per day with a standard deviation of 25 m3 per day.
Once we obtained volume estimates for all unsampled cells,, the area was then divided into
12 daily production blocks (50x50 m) to simulate a harvest operation. To obtain both the
combined predicted sawlog volume and the associated variance estimates for each harvest
block, we again used the methods as described by Journel and Huijbregts (1978) and Kim
and Baafi (1984) to combine the variance estimates from the individual predicted cells for the
12 harvest blocks.
To determine the daily production without the benefits of spatial information, a serpentine
path was placed over the 12 blocks and the daily production levels were obtained by
summing the sawlog volume for all cells within each cutting block. Then, to obtain an optimal
harvest pattern, simulated annealing (Metropolis et al., 1953) was used to find the harvest
pattern that minimized the sum of the squared differences between the demand curve and
the estimated daily production.
Results
The total predicted sawlog volumes for both the non-spatial and spatial sampling were
similar to the actual sawlog volume for the area. The actual total sawlog volume for the area
was 1243.3 m3 with the estimated sawlog volumes for the non-spatial and spatial sampling
methods being 1185.5 m3 and 1196.2 m3, respectively. The variance estimate for the total
predicted sawlog volume from the spatial method was 1067.8 m3. This was less than one
percent of the estimated variance of 74403.4 m3 for the non-spatial method.
The resulting estimated daily sawlog volume production for the two sampling methods
differed as well. Since there was no spatial information for the non-spatial sampling method,
the predicted daily production was simply the total estimated sawlog volume divided by the
total number of operating days. The estimated average daily production from the non-spatial
sampling method was 103.61 m3. The estimated daily production for the spatially explicit
sampling method ranged from a minimum of 69.26 m3 to a maximum of 134.51 m3 and the
actual daily production from the spatial sampling method ranged from 53.61 m3 to a
maximum of 154.01 m3. The sum of the squared deviations between required and produced
volumes for the non-spatial harvest pattern was 7123.27, or more than twice the sum of the
squared differences for the spatially explicit harvest pattern of 3469.15.
Discussion
While the ability to minimize the differences between the consumer’s demand and the
supplier’s production is an important part of any attempt to manage a supply chain, there are
a multitude of issues that prevent the development of a method to minimize the deviation
between the demand and production of log products. Initially, it was assumed a rectangular
sampling grid would yield a sufficient variogram such that little effort would be needed to
obtain the Kriging results. In retrospect, the major task for this study was simply obtaining an
adequate variogram. Once a variogram was obtained, producing a map of the sawlog
volume and resulting variance estimates for the daily cutting blocks was routine with the
software we used.
Lead times that determine optimal delivery for logs will be dictated by transportation
conditions, consumer capacity and storage requirements. By reducing the deviation between
the consumer demand requirements and the estimated production levels, we were able to
demonstrate an advantage of spatially explicit harvest operations. By reducing the lead time
required to deliver logs to the customer, the customer could then reduce or minimizing
storage requirements thus allowing them to focus quality improvements elsewhere and
ultimately improving revenue.
Conclusion
We successfully demonstrated that, while the sample intensity increased, the benefits
associated with being able control harvest operations with better precision outweigh the
additional efforts associated with the additional sampling and processing.
Reference
Brooks, J. R. and Harry V. Wiant, Jr. (2004). Efficient sampling grids for timber cruises.
Northern Journal of Applied Forestry, 21(2):80–82.
Culvenor, D. S. (2002). TIDA: an alogorithm for the delination of tree crowns in high spatial
resoltuion remotely sensed imagery. Computers and Geosciences, 28:33–44.
Dalenius, T., Hájek, J., and Zubrzycki, S. (1961). On plane sampling and related
geometrical problems. In Neyman, J., editor, Proceedings of the Fourth Berkeley Symposium
on Mathematical Statistics and Probability, volume 1, pages 164–177, Berkeley. University of
California Press.
Gambill, C. W., Harry V. Wiant, J., and Yandle, D. O. (1985). Optimum plot size and baf.
Forest Science, 31(3):587–594.
Hay, D. A. and Dahl, P. N. (1984). Strategic and midterm planning of forest-to-product flows.
Interfaces, 14(5):33–43.
Hehnen, M. T., Chou, S. C., Scheurman, H. L., Robinson, G. J., Luken, T. P., and Baker,
D. W. (1984). An integrated decision support and manufacturing control system. Interfaces,
14(5):44–52.
Holmstr¨om, H. (2002). Estimation of single-tree charactersitics using the knn method and
plotwise arial photograph interpretations. Forest Ecology and Management, 167:303–314.
Ihaka, R. and Gentleman, R. (1996). R: A language for data analysis and graphics. Journal
of Computational and Graphical Statistics, 5(3):299–314. Journel, A. and Huijbregts, C. J.
(1978). Mining Geostatistics, chapter 5, page 600. Academic Press, Inc., London.
Kilkki, P. and Päivinen, R. (1987). Reference sample plots to combine field measurements
and satellite data in forest inventory. Helsinginyliopiston mets¨anarvioimistieteen
laitoksentiedonantoja, 19:209–215.
Kim, Y. C. and Baafi, E. Y. (1984). Combining local kirging variances for short-term mine
planning. In Verly, G., David, M., Journel, A., and Marechal, A., editors, Geostistics for
natural resources characterization, volume 122 of Series C: Mathematical and Physical
Sciences, pages 185 199, P.O. Box 17, 3300 AA Dorrecht, Holland. NATO Advanced Study
Institute on Geostistics for natural resources characterization, Reidel Publishing Company.
Korhonen, K. and Kangas, A. (1997). Application of nearest-neightbor regression for
generating sample tree information. Scandinavian Journal of Forest Research, 12:97–101.
Kozak, A., Munro, D. D., and Smith, J. (1968). Taper functions and thier application in
forest inventory. Forestry Chronical, 45:278–283.
Kravchenko, A. N. (2003). Influence of spatial structure on accuracy of interpolation
methods. Soil Science of America Journal, 67:1564–1571.
Mandallez, D. and Ye, R. (1999). Forest inventory with optimal two-phase, two-stage
sampling schemes based on the anticipated variance. Canadian Journal of Forest Research,
29:1691 1708.
Metropolis, N., Rosenbluth, A., Rosebluth, M., Teller, A., and Teller, E. (1953). Equation
of state calculations by fast computing machines. Journal of Chemical Physics, 21:1087–
1092.
Murphy, G., Marshall, H., and Bolding, M. C. (2004). Adaptive control of bucking on
harvesters to meet order book constraints. Forest Products Journal, 54(12):114–121.
Oderwald, R. G. and Jones, E. (1992). Sample sizes for point, double sampling. Canadian
Journal of Forest Research, 22:980–983.
Pebesma, E. (2004). Multivariable geostatistics in s: the gstat package. Computers &
Geosciences, 30:683–691.
Rasinm¨aki, J. and Melkas, T. (2005). A method for estimating tree composition and
volume using harvester data. Scandinavian Journal of Forest Research, 20:85–95.
Stendahl, J. and Dahlin, B. (2002). Possibilities for harvester-based forest inventory in
thinnings. Scandinavian Journal of Forest Research, 17:548–555.
Uusitalo, J. (1997). Pre-harvest Measurement of Pine Stands for Sawing Production
Planning. Acta Forestalia Fennica, 259:56.
Wagner, F. G., Brody, J. A., Ladd, D. S., and Beard, J. S. (1996). Sawtimber valuation and
sawlog allocation through simulation of temple-inland sawmills. Interfaces, 26(6):3–8.
Walthall, C., Kim, M., Williams, D., Meeson, B., Agbu, P., Newcomer, J., and Levine, E.
(1993). Data sets for modeling: a retrospective collection of bidirectional reflectance and
forest ecosystem dynamics multisensor aircraft campaign data sets. Remote Sensing of
Environment, 46:340 346.
Webster, R. and Oliver, M. (2001). Geostatistics for Environmental Scientists, chapter 8,
page 271. John Wiley & Sons, Inc., Baffins Lane, Chichester, West Sussex, PO19 1UD,
England.
Zeide, B. (1980). Plot size optimization. Forest Science, 26(2):251–257.