ENHANCING THE PROFITABILITY OF A VERTICALLY INTEGRATED WOOD PRODUCTS PRODUCTION SYSTEM. PART 2.
HONORIO F. CARINO [*]
DONALD B. WILLIS III [*]
ABSTRACT
This paper presents the results of a case study to demonstrate the efficacy of linear programming in solving a complex set of production-inventory problems frequently faced by integrated wood products manufacturing
Part 1 of this paper [1] dealt with the conceptual aspects and mathematical formulation of a multistage linear programming (LP) model for solving production-inventory problems associated with vertically integrated wood products manufacturing systems. Part 2 of the paper discusses the application of the multistage LP modeling technique in a case study involving an integrated hardwood lumber-cabinet manufacturing operation in the South. The two-part article was developed to demonstrate both theory and practice in generating objective and reliable information for making sound production and inventory decisions in a complex manufacturing environment generally associated with vertically integrated wood products manufacturing operations. For brevity, the reader is referred to Part 1 for more detailed information about the study firm's manufacturing system and operational policies at the time of the study in 1997.
The management of the study firm desired answers to the following questions:
1. What is the optimal (i.e., most profitable) sawlog and lumber production-inventory program over a specified planning horizon (e.g., for each month of the year)? More specifically, what are the types and quantities of logs that should be purchased and processed at the sawmill or kept in inventory per month? What are the types and quantities of lumber that should be kiln-dried and sold or kept in inventory per month?
2. How do changes in the following factors impact system profitability: log procurement and inventory policy, log supply and prices, kiln-dried lumber prices, processing and inventory costs, and manufacturing processes?
This article focuses on the resolution of these issues using the LP model described in Part 1. Based on the mill operating conditions prevailing at the time of the study, the following index sets were used in the actual formulation of the model and in defining the production-inventory system and scope of the problem under consideration:
i = 1 = red oak, 2 white oak, 3 = southern red oak, 4 = poplar, 5 = sweetgum, 6 = hickory, 7 = beech
g = 1 = FAS, 2 = No. 1 Common, 3 = No. 2 Common, 4 = No. 3 Common, 5 = pallet, 6 = cant
d = 8 to 30 inches (1-in. increment)
j = 1 = No. 1, 2 = No. 2, 3 = No. 3, 4 = No. 4, 5 = Miscellaneous
k = 1 = pure oak, 2 = mixed loads (oak, poplar, sweetgum)
p = 1 = January, 2 = February, 3 = March, 4 = April, 5 = May, 6 = June, 7 = July, 8 = August, 9 = September, 10 = October, 11 = November, 12 = December
t = 1 = 4/4-inch, 2 = 3/4-inch
It should be noted that the actual LP model formulated in this case had no upper limits for the lumber sales constraints (i.e., Eq. [16] and [17] in Part 1). This was done to reflect current company policy towards satisfying all the 3/4- and 4/4-inch lumber requirements of its cabinet manufacturing operation first, if possible, before selling some material to outside customers. It was an accurate representation of the fact that there had been no problem in selling any volume of kiln-dried lumber unwanted by the cabinet assembly facility. The company also had no problem selling green lumber to outside customers primarily in the form of cants and pallet materials. Green lumber demand had always exceeded what the sawmill could produce, and the demand of the cabinet plant for kiln-dried lumber had been adequate to keep the existing sawmill operating at capacity.
DATA COLLECTION AND ANALYSIS
LOG INPUT
Historical load tallies of logs procured by the study mill in 1996 and 1997 were utilized in calculating volume distributions of input logs according to species and load type (i.e., pure or mixed) for tree-length logs and species/grade/size combinations for short logs. These tallies were compiled daily using a handheld computer. Mill personnel also kept a tally by species, grade, and size of sawlogs coming from pure and mixed tree-length loads.
There had been a variable quantity of procured logs because of the varying intensities of log delivery experienced by the mill during the year. Log deliveries were often low (1,200 cunits minimum) in the fall (October) and relatively steady and high (up to 2,000 cunits) in the spring (April). Note that I cunit is equivalent to 100 [ft..sup.3] At the time of the study, about 80 percent of the mill's monthly procurement of logs had been in tree-length form and the rest in the form of short cut-to-length logs. Based on volume, the three major species groups (red oak, white oak, and southern red oak) represented almost 80 percent of short logs, 100 percent of pure tree-length loads, and 70 percent of mixed tree-length loads. The volume frequency distribution based on grade classification of short log input was also determined. Furthermore, the volume frequency distributions according to species, grade, and size of sawlogs obtained from straight and mixed tree-length loads were also determined.
Miscellaneous species in short log form were procured in relatively small quantities and not classified according to grade at all. Nevertheless, for identification purposes only, these were given a pseudo grade of No. 4 in the model. Most of the short logs consisted of red, white, and southern red oaks, and more than 70 percent of them graded No. 2 or better.
LUMBER YIELD
A log conversion or yield study at the sawmill was undertaken specifically to determine the lumber yield (volume and grade) of sawlogs from the various types of log input considered. From these data, it was then possible to estimate the volume (cunits) of log input needed to produce a unit volume (1,000 board feet (MBF)) of lumber output.
Per combination of species, size, and grade, up to three sample sawlogs were selected from the different types of log input. For each species/grade combination, sample sawlogs representing 2-inch small-end diameter (SED) increment classes from 8 to 30 inches were selected. However, poplar, sweetgum, hickory, and beech sample log batches were established only on the basis of species and size because they were not graded, as mentioned previously. The samples were randomly pulled out of sawlog piles and measured individually to determine the SED and length classes and then painted at the ends for identification purposes. Each batch had a unique number that identified the log by species, size, and grade. These numbers were painted on the large ends of each log with bright, white paint so that they could be seen in the sawmill during log testing.
In order to minimize error in the gathering of lumber yield data, mill management allowed the test logs to be processed exclusively on two separate Fridays when the mill normally should have been shut down for maintenance. Normal procedures were followed in the processing of individual test logs, but the interarrival time of test logs at the processing centers was deliberately extended to give ample time for the proper identification of each test log, recording of its processing time, and tallying of its lumber yield. At each processing center, one person was assigned to monitor the flow of materials and another to record the necessary data. The test logs grouped by species were processed individually in production runs purposely to eliminate one source of confusion for the grader and tally person.
Regression equations were developed to predict average lumber recovery factor (LRF) values by log species, size (i.e., SED), and/or grade. Table 1 shows the regression relationships that were found to be significant using the stepwise method. SED was a significant factor in each of them, and grade also had a good deal of significance for red oak and southern red oak. As determined previously by other investigators [6], log length was not considered a significant predictor of LRF.
Clearly, in the case of red oak and southern red oak, the LRF value increases with SED up to a certain size (18- to 20-in. SED in this case) where it begins to decrease. This is described as a quadratic relationship and can be attributed to overmature wood in older, larger diameter trees. The larger SED logs come from older trees and the core is often rotten or brashy and yields wood fiber that is brittle or soft and not suitable for lumber [4].
White oak exhibited too much variability so a dependable regression equation could not be developed. Nevertheless, the average LRF values for white oak by size and grade needed for the LP analysis were estimated from the actual mill recovery data.
The prediction equation for miscellaneous species includes a cubic term for SED. This implies that very small SEDs yield high values of LRF that quickly decline and then slightly increase with larger SEDs. This can be attributed to the fact that relatively small-diameter logs of miscellaneous species were converted primarily into cants. Since fewer sawlines were necessary, there were minimum losses in lumber recovery due to kerfs, which are usually large for a circular headsaw like the one used by the study mill. In contrast, red oak and southern red oak logs were converted into 3/4-inch and/or 4/4-inch lumber products only.
Another important set of data needed for the LP analysis was the lumber grade recovery from each test log. Table 2 shows the percentage grade yields of lumber of each species by input log grade.
PROCESSING TIMES
In the LP model, the sawmill is considered as one processing center whose productive capacity is limited by the bottleneck operation. After discussing this issue with the sawmill management, it was apparent that the headrig was the machine center that determined how much lumber could be produced by the sawmill during a regular shift. All the other sawmill operations (debarking, secondary log breakdown or gang sawing, edging, and trimming) were observed to be faster than the headsawing operation. For this reason, the time it took to process each test log at the headrig was determined as a substudy during the log yield study.
The tally person would begin timing when the carriage moved forward for the initial sawline and would stop the timing when the last board or the cant was released from the carriage. Time was recorded in seconds. Any normal delays such as that caused by a hung board were included in the recording of processing times. Obviously, a broken saw tooth caused by hitting a nail was not considered a normal delay and the timing was stopped. In the very few instances when an abnormal delay occurred, the log was discarded from the processing data and not tallied.
Just as for the log yield study, a multiple regression method was used to develop prediction equations (Table 3) for determining the average processing time (in sec.) at the headrig. The same log factors were utilized as predictors. However, all oak species were grouped together to form one regression equation. Similarly, miscellaneous species were grouped together. As expected, the processing time of logs at the headrig increases with SED. In fact, no significant differences existed among oak species or log grades. Among miscellaneous species, only poplar and sweetgum had any significant predictability in terms of processing time. The high predictive ability of the regression equations in Table 3 could be attributed to the fact that there was very little variance in processing time among logs of different grades.
In order to determine the capacity of air-drying and kiln-drying operations, processing times in these operations were obtained from historical data. Company records showed that the air-drying cycles for 3/4- and 4/4-inch red oak, white oak, and southern red oak lumber were about 135 days, and that for 414-inch lumber of miscellaneous species (poplar, hickory, and beech), cycles were 30 days. The kiln-drying cycles for the red oak, white oak, and southern red oak lumber were about 17 to 18 days, and that of the lumber of miscellaneous species was 8 days.
DRYING DEGRADE
A study was also undertaken to determine the amount of degrade occurrence in the drying process. Degrade in small percentages is accepted as a natural element of lumber drying in both air-drying and kiln-drying operations. The company performs both processes at their mill site cognizant of their potential high value-added contribution to the lumber end product.
Due to the time restrictions involved, the air-drying degrade could not be measured, but these data can be easily added to the model when the required input data are available. In addition, only lumber for red oak, white oak, and southern red oak could be tested because the company's miscellaneous species drying inventory was small at the time of the study and none of it was ready to be kiln-dried.
Kiln-drying degrade was estimated based on data obtained from two large packs of air-dried lumber. One pack consisted of lumber graded No. 2 Common and better, and the other pack consisted of lumber graded No. 3 Common. These sample packs were randomly selected from subsets of the stratified population containing packs with both wide and narrow boards. The two packs yielded 3,290 sample boards: 1,704 red oak, 783 white oak, and 803 southern red oak.
The selected packs of lumber were graded, using National Hardwood Lumber Association (NHLA) grading rules [5], prior to and following kiln-drying. Table 4 shows the average percentage of kiln-dried lumber degrade based on volume by species and grade. There was very little drying degrade observed in red oak and southern red oak compared to white oak. Degrade in white oak FAS boards was observed to be relatively high. Other than for FAS, white oak experienced a small percentage of degrade in this operation. Southern red oak exhibited higher drying degrade than red oak due to the extreme amount of cupping and checking inherent in this species.
ECONOMIC AND OTHER MISCELLANEOUS DATA
Recognizing that the information provided by the company is proprietary, an adjustment was made so that the results of the analysis using the economic data provided do not reflect the company's profitability precisely. A separate report that accurately defined the profit and production-inventory figures to be used for planning purposes was submitted to the company. All of the holding, processing, and raw material costs were provided by the company and adjusted for our study. It should be noted that holding costs were calculated as 25 percent of the value of the particular inventory in question. Most companies assess between 15 to 40 percent for the costs of holding inventory from one planning period to the next [2].
The market value for the kiln-dried lumber end products was obtained from weekly issues of the Hardwood Market Report for 1997 [3]. The second week of each month was chosen as the price basis for all lumber species, grades, and sizes. Prices for hickory and beech were provided by the company because the Hardwood Market Report and other publications do not follow their kilndried value closely. The cabinet assembly operation pays market value for lumber that is purchased from the sawmill, and the same price is used for outside sales.
The ending inventory for December 1996 was provided and used as the starting point in the LP analysis. Monthly raw material supply data were obtained from the historical records of the company for 1997. Unit revenue values for green chips, boiler fuel, and pulpwood were also provided by the company. Downtime figures were also provided and calculations were made to determine the average downtime for the sawmill.
LP ANALYSIS
The optimal solution from the base run of the LP model was used to answer the first question posed by the study firm, and the results of sensitivity and parametric analyses [7] were used to answer the second question. The optimization analysis was done over a 12-month planning horizon. However, as stated in Part 1, it was more convenient to use the model structured for a planning period of 1 month and use it repetitively or sequentially in 12 stages to get the same results or inferences for a 12-month planning horizon. This approach proved to be simpler and more convenient in terms of data-set preparation and overall computer modelling or coding, and there were obvious computational advantages.
RESULTS AND DISCUSSION
OPTIMAL PRODUCTION-INVENTORY PROGRAM
Table 5 summarizes the log input volume (cunits) and lumber output volume (MBF) requirements for maximizing net revenue (or profit) of the study mill. These results were derived from the LP base analysis assuming a minimum two-week end-of-month log inventory (i.e., about 600 cunits) and given the existing mill setup and operating and marketing conditions at the time of the study in 1997. The results indicate that the mill has to procure logs amounting to about 14,684 cunits per year or 1,224 cunits per month on average. About 83.1, 3.4, and 13.5 percent of that volume should be in the form of pure tree-length loads, mixed tree-length loads, and cut-up or short logs, respectively. This represents a significant change in log procurement policy because in the past the typical or average log procurement breakdown was 72.6, 7.2, and 20.2 percent for pure tree-length loads, mixed tree-length loads, and short logs, respectively.
As expected, red oak is the predominant species that should be procured. Approximately 82.9 percent of the required annual pure tree-length load volume (12,203.1 cunits), 64.2 percent of the required annual mixed tree-length load volume (496.4 cunits), and 64.8 percent of the required annual short log volume (1,984.7 cunits) should be red oak. The two other major species, white oak and southern red oak, respectively, should be about 9.4 and 7.7 percent of the required annual volume of pure tree-length loads, 11.1 and 8.2 percent of the required annual volume of mixed tree-length loads, and 8.2 and 10.9 percent of the required annual volume of short logs. The predominance of red oak log input in the LP solution actually reflects the company's wood cabinet assembly plant's high demand for and value given to red oak lumber for its internal use. In pure tree-length loads, the predominance of red oak is absolute because it has the same procurement price per ton (or per cunit) as white oak and southern red oak alt hough their lumber volume yields per unit log input volume were found to be not much different. As indicated previously, the volume of white oak and southern red oak that should be procured is not large. In fact, it was determined from the LP base analysis that white oak in pure or mixed tree-length loads should not be procured at all in the months of March, June, July, August, September, and December. In short-log form, white oak should not be procured in the months of January and March. Also, southern red oak in pure tree-length loads should not be procured at all in the months of March, April, and June. In mixed tree-length form, southern red oak should not be procured in March, June, July, August, September, and December. For the most part, such exclusions are directly related to the limitations on the quantity of logs that should be processed and kept in inventory at any given time. Certainly, preference is given to those logs that are less costly to procure and process and/or yield greater product value .
Since mixed tree-length loads were paid a lower price than pure tree-length loads, one would expect that these could provide an economically attractive way to obtain more red oak and southern red oak sawlogs. However, it was found that the quality of sawlogs from mixed tree-length loads was relatively low. Also, mixed tree-length loads were delivered infrequently so that the available supply of sawlogs from them had been small most of the time. This helps explain the low procurement levels of mixed tree-length loads shown in Table 5. In fact, mixed tree-length loads with red oak as the predominant species and with a significant volume of poplar should be procured in January, February, April, May, October, and November only. Instead of mixed tree-length loads, pure tree-length loads of red oak should be procured for the rest of the year.
It was also determined that in the case of red oak short logs, procurement efforts should be focused on No. 1 and No. 2 grades, which represent about 65 and 25 percent of the total required annual red oak short log procurement volume. Surprisingly, grade No. 4 red oak short logs should be procured much more than No. 3 grade. This could be explained by the findings from the recovery study that showed that grade No. 4 short logs, which were smaller in size but sounder in quality and cheaper, had relatively higher LRF values than No. 3 grade short logs. In the case of white oak short logs, the firm should procure grade No. 3 short logs almost exclusively except in the months of October and November when a small volume of grade No. 1 short logs must be procured, mainly to satisfy log inventory requirements. This can be explained simply by the fact that No. 1, No. 2, and No. 3 white oak short logs were found to have no significant differences in lumber yield, and the grade No. 3 log is cheaper. In the case of sou thern red oak short logs, most of the procured volume should be in the form of grades No. 1 (52%) and No. 2 (32%). Like red oak, grade No. 1 southern red oak short logs have been procured by the mill whenever available. However, the mill should only procure small amounts of No. 3 and No. 4 southern red oak short logs, i.e., about 1.8 and 1.2 cunits per month on average, respectively.
The results of the LP base analysis also indicate that the procurement of relatively small quantities of miscellaneous species such as poplar, hickory, and beech is necessary. Hickory and beech, which have been available in short log form only, should represent about 11.7 and 2.9 percent of the short log volume required, respectively. Poplar, which should represent about 16.4 percent of the mixed tree-length volume and 1.5 percent of the short log volume, proved not to be needed by the mill most times of the year. The company might want to look at procuring poplar logs in tree-length form exclusively. This is in direct contrast to the procurement decisions at the time of the study; the mill was then opting to purchase all available poplar short logs. Furthermore, the procurement of sweetgum in any form should be stopped completely because the LP model has clearly identified this to be submarginal, i.e., it has a negative impact on mill profit. Heretofore, the mill had been converting sweetgum logs into low-v alued pallet materials only.
The required log procurement volume changes by month due to the inherent variability in log supply. According to management, fall and early winter months are the times when log quality and supply suffer the most during a normal year. Therefore, log inventory has to be built up during the late summer months. The results of the LP base analysis suggest that the mill has to increase log procurement efforts in the months of October and November, and might even have to procure more logs of the less profitable species such as white oak and poplar, in order to have sufficient inventory for winter operations.
Table 5 also shows the optimal volume of kiln-dried lumber by size, pallet lumber, and cant lumber that the study mill should produce and sell per month as derived from the LP base run solution. In order to maximize profit, the mill with its present setup evidently has to produce and sell approximately 6 million board feet of lumber per year, the bulk of which consists of kiln-dried red oak lumber. On a thickness basis, red oak should represent about 81.3 percent of the total volume (4,267.4 MBF/yr.) of 4/4-inch and 100 percent of 3/4-inch (275.9 MBF/yr.) kiln-dried lumber sales volume. If southern red oak were added to the red oak kiln-dried lumber sales, it would represent approximately 90 percent of the total volume of kiln-dried lumber sold. This is higher than the 80 percent level experienced on a regular basis in the past, as mentioned in Part 1. This difference can be attributed to the more efficient allocation of log input as suggested by the model. Such allocation does not include as much white oak and miscellaneous species volume as what was being processed by the mill at the time of the study. Only about 5.1, 8.0, 0.9, 3.4, and 1.2 percent of the total volume of 4/4-inch kiln-dried lumber produced and sold should be white oak, southern red oak, poplar, hickory, and beech, respectively.
It was also determined that pallets and cants should represent about 22.7 (1,358.1 MBF) and 1.5 percent (89.1 MBF) of the mill's total annual sales of lumber, respectively. Clearly, very little cant volume should be produced from predominantly white oak logs. It is projected that much of this cant production would occur in October and November, when less profitable logs of white oak and poplar are expected to be procured in relatively higher volumes.
RESULTS OF SENSITIVITY ANALYSIS
Sensitivity analysis was conducted to determine the range of values within which the LP solution remains optimal. Of major interest to us were: 1) the reduced costs or dual values associated with objective function coefficients of decision variables that are not part of the optimal solution; and 2) the shadow prices or dual values associated with the right-hand side of resource constraints that are limiting (i.e., no slack). The following discussions on sensitivity analysis and the subsequent parametric ("what if') analysis are presented herein using the results of the LP base run for the month of June. It exemplifies a period when steady-state conditions have been achieved, allowing us to represent an average monthly operation.
Reduced cost is the reduction in net revenue that would result from continuing to procure submarginal sawlogs. It is equivalent to the amount that could be added as maximum premium to the price of logs that the LP model considers optimal to process, with the end view of securing sufficient supply of such logs. Conversely, it is equivalent to the minimum amount that prices of submarginal logs (per the LP model) should be reduced to, before such logs could be procured by the mill without diminishing its current profitability.
Table 6 shows the weighted (by probability distribution of SED) average reduced costs for short logs. Red oak grades No. 2 and No. 3 short logs had sizes that did not enter the optimal solution, and hence had associated reduced costs. If the mill continued to procure No. 2 grade short logs less than 12 inches in diameter, a reduction in net revenue amounting to at least $10.65 per cunit purchased would be expected. Procurement of grade No. 3 short logs less than 10 inches in diameter would reduce net revenue by at least $14.74 per cunit on average. This brings up an interesting point. At the time of the study, the mill had not been processing sawlogs with SEDs less than 8 inches. Looking at all the species with reduced costs here, it would seem that the mill might substantially enhance its net revenue by establishing a lower limit of 12 inches for SED.
Another important insight is that in order to be considered feasible for processing at the mill, poplar short logs and grade No. 1 white oak short logs prices must be reduced by at least $1 6.75/cunit and $25.99/cunit, respectively.
In the case of tree-length loads, only white oak logs in pure tree-length loads and southern red oak, poplar, and sweet-gum logs in mixed tree-length loads had reduced costs, which were about $5.26/cunit on average. Again, this means that the mill could consider processing these types of log input if the price were lowered by at least $5.26 per cunit. This amount is equivalent to how much premium the mill could afford to pay for pure tree-length loads of southern red oak or red oak in order to encourage delivery of the required quantities of these materials.
Shadow price is defined as the value by which the LP objective function increases if one more unit of resource or capacity can be added to the right-hand side of constraints that are limiting. Like reduced costs, shadow prices can be looked at as a premium to offer in log procurement pricing in order to obtain the desirable type and quantity of logs. Unlike reduced costs, however, shadow or dual prices are associated with those decision variables that enter the solution.
Table 7 shows the weighted average shadow prices ($/cunit) for short logs by species and grade. Evidently, net revenue would increase by as much as $114.25 per additional cunit of No. 1 red oak short logs processed. In other words, up to $114.25 of premium could be paid per additional cunit of such logs to guarantee their delivery and without expecting any change in the current optimal solution. Also noteworthy is the comparison between grade No. 1 red oak and grade No. 1 southern red oak short logs. The log and lumber prices for both species were the same. However, the estimated maximum premium ($114.25/cunit) that could be offered for red oak short logs is more than four times that of southern red oak ($25.36/cunit). This could be attributed to the relatively better lumber volume and grade yields of red oak as mentioned earlier. It is also clear from Table 7 that in order to increase the delivery of hard-to-get logs of beech and hickory, the mill could offer a maximum premium of $80.18 and $41.11, respecti vely, per additional cunit of such logs.
Shadow prices obtained for tree-length log input, particularly pure red oak tree-length loads, also suggest that a maximum premium of about $40/cunit could be offered to suppliers in order to encourage the delivery of the desired quantity of that material. There were no shadow prices, hence no premium, for the other species in tree-length log form because the supply or resource constraints associated with such species were not limiting (i.e., slacks or oversupply existed).
The shadow prices associated with sawmill, air-drying shed, dry kiln, and dry warehouse capacity constraints were also estimated to determine how net revenue would be impacted by changes incapacity. All of the capacities at each of these processing centers except the sawmill exhibited slack and therefore had no shadow prices. In fact, only about one-third of the available capacity at the air-drying sheds and kilns was used. Therefore, it is safe to say that no capacity needs to be added at those operations.
Sawmill capacity had a shadow price of about $4.29, which could be interpreted as the expected enhancement of net revenue for every minute increase in productive time at the sawmill. Sawmill productive time could be increased by adding a second shift or paying overtime wages. The shadow price suggests that the mill could maintain the same optimal level of profitability even if it had to pay up to $4.29 per additional minute increase in sawmill productive time.
RESULTS OF PARAMETRIC ANALYSIS
Parametric or "what if" analysis was conducted to address Question 2: How do changes in the following factors impact system profitability: log procurement and inventory policy, log supply and prices, kiln-dried lumber prices, processing and inventory costs, and manufacturing processes?
It was determined that the company could realize an average increase of 156 percent in net revenue per month by changing the current log inventory policy from a 1-month level to a 2-week level. That is, it is more economical to maintain a smaller amount of log inventory that would allow the mill to run continuously for 2 weeks at most before log supplies are again replenished. In this connection, the company should also improve its log pricing strategy so that only the proper or desired species, grades, and sizes of logs are purchased. As stated previously, offering price premium incentives for what the mill needs could help control the type and level of log inventory.
Another procurement policy change that might be considered by the mill relates to the minimum SED of short logs or top diameter of tree-length logs. At the time of the study, 8 inches was the minimum SED or top diameter tolerated. Results of parametric analysis indicate that increasing the SED or top diameter to 10, 12, and 14 inches would result in percentage increases in net revenue of about 5.2, 11.7, and 15.5 percent, respectively. Apparently, the most attractive policy is to procure logs with a minimum SED or top diameter of 12 inches. Changing the minimum SED or top diameter from 10 to 12 inches could result in more than a doubling of the percentage change in net revenue. However, increasing the minimum SED or top diameter level to 14 inches could result in an increase of only 3.8 percentage points over the 12-inch level. It should be noted, however, that increasing the minimum SED or top diameter could discourage many suppliers from delivering logs to the mill unless premium price incentives are given.
Changes in the supply quantity of short logs appeared to have no significant impact on mill profitability. It was determined that for every percent increase in the supply of No. 1 red oak short logs, the mill could realize a net revenue increase of approximately only 0.1 percent. Since the estimated net revenue in June was about $58,657 and there were about 70 cunits of No. 1 red oak short logs processed, that translates to a mere $84 increase in net revenue per cunit increase in that type of log input. As a matter of fact, the impact on profitability of other grade categories of red oak short logs was found to be negligible. Also, changes in the log supply quantity of miscellaneous species had literally no effect on mill profitability. This was expected because they represent a very small proportion of the total log input volume required, and hence are considered not vital to the performance of the company. In contrast, pure tree-length load supply was observed to have a large impact on net revenue. A 10 pe rcent increase in pure tree-length load supply would result in net revenue improvement of about 5.7 percent for the month of June. On the other hand, a 10 percent decrease in pure tree-length load supply would result in net revenue reduction of about 6.6 percent.
Log procurement prices also have a considerable impact on mill profitability. For example, if prices of all (i.e., regardless of species and grade) short logs procured decreased by 10 percent, system profitability would increase by as much as 16 percent. On the other hand, system profitability would decrease by 32 percent with a 10 percent increase in short log prices. As expected, changes in the price of the two most important species (red oak and southern red oak) have a significant impact on system profitability. For example, a 10 percent reduction in the price of red oak and southern red oak short log inputs would result in a net revenue increase of about 5.6 percent. Note that this is about one-third of the percent increase in net revenue expected from a 10 percent reduction in price of all short logs. This again confirms the strategic importance of red oak and southern red oak to the mill operation.
Significant value is added when lumber is kiln-dried and sold versus selling the lumber green or even air-dried. Due to the high profit margins from kiln-dried hardwood lumber, changes in prices for this product can have a tremendous impact on net revenue attainable. For example, a 5 percent increase in the prices of all kiln-dried lumber sold would result in about a 25.5 percent increase in net revenue. Almost 95 percent of this net revenue increase could be attributed to red oak and southern red oak lumber, once again underscoring the strategic importance of these species. On the other hand, changes in lumber prices for white oak had virtually no effect on system profitability. This further enhances the view that white oak is not considered important to the operation as asserted earlier.
Changes in processing and inventory holding costs definitely have a major impact on system profitability. For example, it was determined that a 10 percent increase in sawmilling and log holding costs would result in net revenue reductions of about 21 and 6.2 percent, respectively. On the other hand, a 10 percent decrease of the same cost items would result in net revenue increases of 17.9 and 3.1 percent, respectively. Interestingly, the percent increase in net revenue quadruples when holding cost was decreased by 20 percent. This helps explain why the company should change from a 1-month to a 2-week log inventory policy as suggested earlier.
As mentioned previously, the productive capacity of the vertically integrated production system is limited or determined by the sawmill. It was determined that adding a second work shift would result in a 23 percent increase in net revenue, certainly an attractive option that the sawmill manager must consider.
System profitability is also greatly impacted by improvements in lumber yield and drying degrade percentage. For example, it was found that a 6 percent increase in lumber volume yield would result in a net revenue increase of about 20.2 percent. However, system profitability seems to be much more sensitive to reduction in lumber drying degrades. A 6 percent reduction in lumber drying degrades would result in a net revenue increase of about 31.3 percent.
SUMMARY AND CONCLUSIONS
LP was used to analyze a complex set of production-inventory problems commonly faced by integrated wood products manufacturing operations exemplified by the study mill, which is a vertically integrated hardwood lumber-cabinet manufacturing operation in the South. Specifically, this study was undertaken primarily to determine the optimal (i.e., most profitable) production-inventory program that the lumber manufacturing sector of the study mill could have over a specified planning horizon (e.g., for each month of the year). This involved determining on a monthly basis the most profitable combination of the types and quantity of logs that should be purchased and processed at the sawmill or kept in inventory and the types and quantity of lumber that should be produced and sold or kept in inventory. Also, the impact of changes in log procurement and inventory policy, log supply and prices, kiln-dried lumber prices, processing and inventory costs, and manufacturing processes on system profitability was evaluated t hrough sensitivity and parametric analyses.
The following conclusions could be drawn from results of the LP analysis with the objective of maximizing net revenue or profit, given the existing mill setup and operating and marketing conditions at the time of the study:
1. The annual log procurement level should be set to about 14,684.2 cunits (i.e., 1,224 cunits per mo. on average), about 83.1, 3.4, and 13.5 percent of which should be in the form of pure tree-length loads, mixed tree-length loads, and cut-up or short logs, respectively. Red oak should be procured as the predominant species representing about 82.9, 64.2, and 64.8 percent of pure tree-length loads, mixed tree-length loads, and short logs required, respectively. In the case of short logs, the focus should be in the procurement of red oak logs graded No. 1 and No. 2, which represent about 65 and 25 percent of the total required volume, respectively. In order to increase the likelihood that the required volume of red oak No. 1 grade short logs would be delivered, a premium of up to $114.25/cunit could be extended to suppliers. Similarly, a premium of up to $40/cunit of red oak pure tree-length loads could be extended because these were found to yield quality red oak saw-logs also.
2. The sawmill has to operate at capacity and produce about 6 million board feet of lumber per year (or 500 MBF per mo.), about 71.2 and 4.6 percent of which should be sold as 4/4- and 3/4-inch kiln-dried lumber, respectively; the remainder, 22.7 and 1.5 percent, should be sold as green pallet materials and cants, respectively. Red oak and southern red oak were found to represent over 90 percent of the total annual kiln-dried lumber production or sales volume. This underscores the strategic importance of these two major species to the cabinet assembly plant, which had been depending upon the company-owned sawmill to meet all its red oak and southern red oak lumber needs.
3. The mill should no longer consider procuring sweetgum in any form and poplar short logs because they were found to be unprofitable to process. Poplar should be considered acceptable only if the price were reduced by at least $16.75 per cunit for short logs and $5.26 per cunit for mixed tree-length loads.
4. A log procurement policy that would allow the study mill to keep at least a 2-week inventory (600 cunits) of logs should be adopted because this could result in profit improvement of up to 156 percent, using the minimum 1-month log inventory policy at the time of the study as a basis.
5. The mill should consider imposing a 12-inch minimum SED level for all logs delivered to the mill because this would improve net revenue significantly. For example, an estimated 11.7 percent increase in net revenue in the month of June could be realized with the 12-inch minimum SED.
6. Due to the scarcity of highly valued beech short logs, a price premium of up to $80.18/cunit could be offered to encourage the delivery of these logs by suppliers, including those outside the current procurement region.
7. The white oak short log procurement policy should be reexamined because it was determined that the mill essentially needs to purchase No. 3 grade logs only. The company should consider not buying white oak in tree-length form anymore because a sufficient supply of white oak short logs had been available most of the time.
8. It was determined that the mill could afford to pay up to $4.29 per additional minute increase in productive time, which could be accomplished by working overtime or adding another shift. The results of the analysis show that adding a second work shift seems to be an attractive option because it could result in net revenue increases of about 23 percent.
9. Mill profitability was found to be very sensitive to changes in kiln-dried lumber prices, sawmill conversion efficiency, and lumber drying degrades; moderately sensitive to changes in log supply (particularly tree-length logs) and prices, processing costs, and inventory costs; and insensitive to changes in the supply quantity of short logs.
The authors are, respectively, an Associate Professor and a former graduate student, School of Forestry and Alabama Agri. Expt. Sta., Auburn Univ., Auburn, AL 36849. This paper is published as Alabama Agri. Expt. J. Series 9-985955. This paper was received for publication in July 2000. Reprint No. 9144.
(*.) Forest Products Society Member.
LITERATURE CITED
(1.) Carino, H.F. and D.B. Willis III. 2001. Enhancing the profitability of a vertically integrated wood products production system. Part 1. A multistage modelling approach. Forest Prod. J. 51(4):37-44.
(2.) Fogarty, D.W., J.H. Blackstone, and T.R. Hoffmann. 1991. Production and Inventory Management. 2nd ed. South-Western Pub. Co., Cincinati, OH. 870 pp.
(3.) Hardwood Market Report. 1997. Lumber News Letter. Hardwood Market Rept., Memphis, TN.
(4.) Lane, P.H., J.W. Henley, R.O. Woodfin, and M.E. Plank. 1973. Lumber recovery from old-growth coast Douglas-fir. Res. Pap. PNW-154. USDA Forest Serv., Pacific Northwest Forest and Range Expt. Sta., Portland, OR. pp.
(5.) National Hardwood Lumber Association. 1986. Rules for the measurement and inspection of hardwoods and cypress. NHLA, Memphis, TN. 118 pp.
(6.) Steele, P.H. 1984. Factors determining lumber recovery in sawmilling. Gen. Tech. Rept. FPL-39. USDA Forest Serv., Forest Prod. Lab., Madison, WI. 8 pp.
(7.) Vanderbei, R.J. 1996. Linear Programming: Foundations and Extensions. Kluwer Academic Pub., Boston/London/Dordrecht. pp. 101-113.
Prediction equations for determining
average LRF by log species, SED, and/or grade.
Species Prediction equation [a] [r.sup.2]
Red oak LRF = 2.8300846 + 0.4143348 x (SED) - .726
0.0103 x ([SED.sup.2] - 0.477357 x
(Grade #2) - 0.640487 x (Grade #3)
Southern red oak LRF = 4.136097 + 0.2061389 x (SED) - .735
0.00561 x ([SED.sup.2]) + 0.448167 x
(Grade #1)
Miscellaneous LRF = 27.151853 - 3.850489 x (SED) + .819
0.21785 x ([SED.sup.2]) - 0.003836 x
([SED.sup.3]) + 0.3552135 x (HIK)
(a.)Where: LRF = Lumber recovery factor in board feet of lumber output per cubic feet of log input (BF/[ft..sup.3]); SED = Small-end diameter of saw log (in.); Grade #1 = Sawlog of Grade No. 1 (if yes = 1, if no = 0); Grade #2 = Sawlog of Grade No. 2 (if yes = 1, if no = 0); Grade #3 = Sawlog of Grade No. 3 (if yes = 1, if no = 0); HIK = Hickory sawlog (if yes = 1, if no = 0).
Percentage lumber grade yield by input
log grade for various species.
Lumber grade yield
Log species 4/4 thickness
and grade FAS No. 1C [a] No. 2C No. 3C
(%)
Red oak
No. 1 25.3 42.2 7.3 11.9
No. 2 3.2 30.4 18.7 21.7
No. 3 0.0 15.9 10.1 28.3
No. 4 0.0 11.7 12.6 33.0
White oak
No. 1 17.0 31.4 17.8 8.3
No. 2 0.6 21.5 17.5 23.4
No. 3 1.3 22.5 23.5 19.1
Southern red oak
No. 1 16.7 43.4 11.9 9.1
No. 2 5.6 33.7 21.0 14.2
No. 3 0.9 18.9 21.1 14.1
No. 4 3.5 14.0 5.5 18.7
Poplar 18.9 38.4 14.4 15.6
Sweetgum 0.0 0.0 0.0 0.0
Hickory 7.4 34.5 15.5 23.1
Beech 14.4 38.4 15.1 11.7
Log species 3/4 thickness
and grade Pallet FAS No. 1C No. 2C No. 3C
Red oak
No. 1 10.6 0.8 0.7 0.6 0.5
No. 2 22.7 0.7 0.7 0.8 0.7
No. 3 43.7 0.0 0.7 0.7 0.3
No. 4 41.4 0.0 0.0 0.0 0.0
White oak
No. 1
No. 2
No. 3
Southern red oak
No. 1
No. 2
No. 3
No. 4
Poplar
Sweetgum
Hickory
Beech
Log species
and grade Pallet Cant
Red oak
No. 1 0.1
No. 2 0.4
No. 3 0.3
No. 4 1.3
White oak
No. 1 13.5 12.0
No. 2 18.8 18.2
No. 3 17.2 16.4
Southern red oak
No. 1 18.9
No. 2 25.5
No. 3 45.0
No. 4 58.3
Poplar 0.2 12.5
Sweetgum 83.0 17.0
Hickory 5.4 14.1
Beech 3.4 17.0
(a.)C = Common.
Prediction equations for the
determination of average processing time
(sec.) at the headrig by species, SED, and grade.
Species Prediction equation [a] [r.sup.2]
Oak PT = 145.24769 - 21.76372 x (SED) +
1.102218 x ([SED.sup.2]) .929
Miscellaneous PT = 274.63154 - 37.30394 x (SED) +
1.481681 x ([SED.sup.2]) - 10.42403
x (POP) - 16.97055 x (GUM) .976
(a.)Where: PT = processing time at the
headrig (sec.); SED = small-end diameter
of sawlog (in.); POP = poplar sawlog (if
yes = 1, if no = 0); GUM = Sweetgum
sawlog (if yes = 1, if no = 0).
The average percentage of kiln-dried
lumber degrade based on volume by
species and grade.
Lumber grade
Species FAS No. 1C [a] No. 2C No. 3C
(drying degrade, %)
Red oak 2.2 1.2 2.0 5.7
White oak 14.9 4.5 4.7 0.0
Southern red oak 4.8 4.6 4.5 3.2
(a.)C = Common.
Required log input volume (cunits) and
lumber output volume (MBF) for maximizing
net revenue (or profit) of the study mill
based on the LP base run (i.e., given the
existing setup and operating and market
conditions at the time of the study in 1997).
Log input volume
Procured logs
Tree-length loads
Sawlogs
Month Pure Mixed Short logs Total processed
(cunits)
Jan. 867.3 108.4 66.5 1,042.2 865.7
Feb. 1,097.2 10.8 114.6 1,222.6 877.5
Mar. 1,149.9 38.1 123.5 1,311.5 926.0
Apr. 1,134.2 47.2 165.7 1,347.1 977.4
May 957.2 107.8 181.1 1,246.1 909.4
Jun. 1,034.1 57.7 165.1 1,256.9 920.2
Jul. 988.0 26.5 183.1 1,197.6 886.8
Aug. 1,066.1 23.9 139.6 1,229.6 885.1
Sep. 1,108.5 31.8 192.7 1,333.0 990.6
Oct. 908.3 24.7 297.4 1,230.4 934.3
Nov. 1,005.5 14.5 183.6 1,203.6 887.5
Dec. 886.8 4.9 171.8 1,063.6 779.7
Total 12,203.1 496.4 1,984.7 14,684.2 10,840.2
Lumber output volume
Kiln-dried lumber Green lumber
Month 4/4-inch 3/4-inch Pallets Cants Total
(MBF)
Jan. 474.5 113.5 109.1 7.5 704.6
Feb. 333.9 14.8 106.3 13.6 468.6
Mar. 352.5 18.1 112.4 3.6 486.6
Apr. 370.3 17.1 122.9 5.2 515.5
May 343.2 13.1 118.1 3.7 478.1
Jun. 346.3 14.7 119.0 4.5 484.5
Jul. 341.4 16.1 111.2 3.4 472.1
Aug. 338.1 17.0 110.2 2.1 467.4
Sep. 375.1 16.6 130.1 2.6 524.4
Oct. 361.6 10.2 111.9 21.3 505.0
Nov. 327.5 10.5 109.1 20.1 467.2
Dec. 303.0 14.2 97.8 1.5 416.5
Total 4,267.4 275.9 1,358.1 89.1 5,990.5
Weighted (by probability distribution of
SED) average reduced costs or reduction
is revenue ($/cunit) for allowing the
processing of submarginal short logs of
specified species, grade, and diameter ranges.
Small-end
Species Log grade diameter range
(in.)
Red oak No. 2 Less than 12
Greater than 23
No. 3 Less than 10
White oak No. 1 All sizes
No. 2 Less than 12
No. 3 Less than 12
Greater than 18
Southern red oak No. 1 Greater than 25
No. 2 Less than 11
Greater than 21
No. 3 Less than 12
Greater than 17
Poplar Miscellaneous All sizes
Reduced cost
Species (or reduction in revenue)
$/cunit)
Red oak 10.65
8.31
14.74
White oak 25.99
16.28
6.88
9.81
Southern red oak 19.17
3.26
13.39
6.51
5.51
Poplar 16.75
Weighted average shadow price ($/cunit)
for short logs by species and grade.
Log grade
Species No. 1 No. 2 No. 3 No. 4 Miscellaneous
($/cunit)
Red oak 114.25 8.80 18.41
White oak 11.06
Southern red oak 25.36 6.58 1.01 12.66
Poplar
Sweetgum
Hickory 41.11
Beech 80.18
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