Assessing the effectiveness of static heuristics for scheduling lumber orders in the sawmilling production process

: Although optimization models can be used to plan the production process, in most cases static heuristics, such as earliest due date (E), longest processing time (L), and shortest processing time (S), are used because of their simplicity. This study aims to analyze the production cost of the static heuristics and to determine how this cost relates to the size of the production orders in the sawmilling industry. We set a planning problem with different orders and due dates and solved it using two cost-minimization models to compare their solutions. The first was a planning model (PL) where orders were split up into products demand by period, and the second, a planning scheduling (PS) where the sequence of processing orders based on static heuristics was assumed as known. In the latter, the minimum production cost for each static heuristic was found. In both models, the same resource constraints were assumed. The costs showed no significant changes based on order sizes. However, 0,5 % of orders were delayed using PS-E, and 17 % of orders were delayed using PL. PL was an efficient solution method when changing the orders´ size and when looking for the best static heuristic to process the orders. However, PS-E showed the ability to reduce the backlog close to zero while the PL backlog ratio was 17 %. No penalties were applied to backlogs due to their subjective nature; however, when shortages occurred, the demand was unmet or backlogged with substantial costs. Thus, in case the proposed method is adopted using a conservative backlog cost, a sawmill producing under the cut-to-order environment that produces 300000 m 3 /year would reduce backlogged orders by 51000 m 3 . If the holding lumber cost is 2 $/m 3 , annual savings would be $408000.


Introduction
In the year 2020, the forest industry emerged as one of the key pillars of the Chilean economy, contributing 2,1% to the national GDP.Forty percent of all industrial roundwood was used in sawmills, with a similar percentage allocated to pulpwood and paper production.The remaining portion was used in the manufacturing of lumber, panels, and other products, many of which were exported.Chilean forest companies exhibit a high level of integration, encompassing operations in forestry, cellulose, paper, lumber, plywood, and energy production.Nevertheless, there has been a growing focus on lumber production, driven by the need to address significant efficiency gaps.
In most large sawmills in Chile, the planner faces the task of balancing the utilization of logs.
This involves assigning sawing patterns to logs to meet orders deadlines while ensuring a certain performance threshold regardless of its impact on production costs.Unfortunately, these performance thresholds often clash with the efficient utilization of capacity.For instance, this conflict arises when scheduling small batches for numerous orders without considering setup costs.Conversely, scheduling large batches for a few orders prioritizes maximum productivity.This paper seeks to quantify the impact of static heuristics for scheduling, widely employed by sawmill planners, and the influence of lumber batch size on production costs.The objective is to identify the production setting in which static heuristics outperform others.This section begins with a literature review, highlighting gaps in existing research, and establishes the research objectives.
Lumber planning is based on a balance between timber supply and lumber demand.Sawmills that process timber from natural forests typically employ the cut-to-stock approach, whereas those processing timber from plantations favor the cut-to-order approach (Vanzetti et al. 2018, Marier et al. 2014).The Operational Lumber Planning Problem (OLPP) can be categorized as both a lot sizing and scheduling problem, determining when orders should be fulfilled while taking capacity into account to minimize backlog and inventory costs.
The OLPP has been approached through Mixed Integer Programming (MIP) formulations, heuristics methods, and hybrid approaches that combine the strengths of both.MIP models play a crucial role in minimizing the costs associated with raw materials, inventory, and backlog while adhering to available capacity constraints (Clark 2003).However, scheduling presents challenges due to sequence-dependent setups, where each machine's setup time for a task is influenced by both the current job and the previous job for which the machine is setup.Additionally, multiple setups occur within a planning period.As a result, the MIP model for lot sizing and scheduling involves numerous binary variables, leading to computational intractability.While exact formulations produce high-quality solutions, they often come with impractical solution times.To overcome this complexity, a strategy involves relaxing the binary variables and constraints, breaking down the problem into smaller components to determine lot size, and establishing a sequence of lot setups (de Araujo et al.

and Clark 2003).
Maness and Adams 1991, followed by Maness and Norton 2002, conducted early research on lumber planning and scheduling.The MIP model they developed aimed to maximize revenues by generating a bucking-sawing pattern and selecting the mix of logs to meet sales demand.This approach is suitable when backorders and inventories are allowed.Sawmills commonly use MIP models with a focus on either volume or value.However, if dynamic prices can be applied and updated during sawing operations, optimized plans can yield more accurate plans resulting in using fewer logs (Todoroki and Ronnqvist 2010).
Adopting a cut-to-order environment, where logs are sorted by diameter and processed in batches, has improved the results of the OLPP through timber transfer decisions among plants within a supply chain context (Singer and Donoso 2007).A heuristic approach mimicking planner schedulers and a MIP model were developed and benchmarked by Huka and Gronalt 2017.The MIP model outperforms the heuristics, however, when applying the MIP model in a rolling planning horizon, the best heuristic outperforms it owing to the end effects included in the model.
An alternative approach was undertaken by Dumetz et al. 2019 In a cut-to-stock environment, Wery et al. 2014 emphasized the challenges that arise when dealing with customized orders.They proposed a simulation-optimization sequential framework to evaluate how mill settings and the introduction of new log classes impact the sawmill products mix when customized products are added to the plan during the planning horizon.Furthermore, Ben Ali et al. 2019 outlined the advantages of integrating sales and operations planning and order promising based on revenue management concepts.Their MIP model determines sales decisions that maximizes profit while enhancing service levels for high-priority customer orders.
In a cut-to-order environment, incorporating remanufacturing operations, Vanzetti et al. 2019 developed a MIP model for day-ahead scheduling optimization.This model utilizes both selfgenerated and data-based cutting patterns, emphasizing the advantage of generating all possible cutting patterns, which surpasses previous research efforts.
To address the OLPP in a more general context, Kaltenbrunner et al. 2020 integrated existing approaches and evolving scenarios.They developed a modular, flexible, and generic planning method, with MIP model cases differing in demand fulfillment constraints, optimal use of raw material constraints, and the division of the planning horizon.
While previous reviews have focused on MIP and heuristic methods, this research aims to quantify the impact of static scheduling heuristics on production costs.Therefore, a subsequent review of scheduling methods applicable to this initiative follows.
Scheduling is typically conducted by shift leaders without considering cost consequences.A schedule arranges jobs in a specific order and determines their start and completion times (Maccarthy and Liu 1993).Operations Research (OR) methods can be employed to address scheduling problems, aligning with the production flow and production system.Different objective functions, indirect objectives, and metrics can be produced based on the problem scale, allowing for the determination of solution technique performance gaps (for further details, refer to Maccarthy and Liu 1993).While the optimal schedule serves as a performance measure, it is important to note that costs and profits cannot be directly linked to schedules.Indirect objectives, such as completion time, flow time, lateness, and tardiness are used instead (Sipper and Bulfin 1997).
OR methods employed to tackle scheduling problems encompass optimal methods, enumerative optimal MIP methods, and heuristic methods, the latter involving a certain degree of closeness to polynomial time (Maccarthy and Liu 1993).Additionally, static heuristics offers optimal solutions for certain problems, and they are a function of the order data (Possani 2001).In fact, static heuristics prioritize jobs and schedules them based on a rule that recognizes the priorities of all the orders that are waiting to be processed on a machine.The Shortest Processing Time (SPT) static heuristic prioritizes jobs based on their processing times, while Earliest Due Dates (EDD) static heuristic prioritizes jobs based on their due dates.The Longest Processing Times (LPT) static heuristic organizes jobs in a sequence with non-increasing processing times.Unfortunately, neither SPT nor LPT considers due dates, as these static heuristics primarily focus on minimizing flow time (i.e., cost minimization).If customer satisfaction is a crucial factor, EDD is the preferred choice because it minimizes tardiness (Sipper and Bulfin 1997).Yaghubian et al. 2001 developed a model to address a dry kiln scheduling problem, a version of scheduling "n" jobs on "m" parallel machines.Jobs were assigned to kilns in accordance with their due dates, and the optimization problem revolved around allocating jobs to kilns.
Each job was associated with demands (e.g., orders), and the due date was ensured by imposing a constraint that guaranteed the fulfillment of the last job on demand by the due date.
Furthermore, Maturana et al. 2010 tackled the OLPP using a Linear Programming (LP) model and a heuristic approach that emulated planner decisions.They introduced log supply and demand perturbations.While the LP model proved to be faster than the heuristic approach for most scenario perturbations, it lacked explicit scheduling rules present in the heuristic method, such as LPT.In a similar vein, Pradenas et al. 2004  This simplification aimed to reduce complexity and make the problem more tractable (Vanzetti et al. 2019).
Although efforts have been made to address the OLPP, much of the research has prioritized the due date of orders over production cost.Limited attention has been given to understanding how static heuristics for scheduling can impact costs.This research quantifies the effect on cost when using static heuristics for scheduling lumber orders on sawmills.
The following sections provide a methodology outlining the approach, model formulations, data utilized, and the comparison strategy.Subsequently, the results are presented and benchmarked against previous research, followed by a detailed discussion of findings and research limitations.We conclude with summaries of conclusions, implications, and study limitations.

Materials and methods
The methodology was designed to examine the impact of incorporating static heuristics, such as EDD, SPT, or LPT, in processing lumber production orders on cost, backlog, and overdue orders while solving the OLPP.The formulated models designed to address this problem are described, along with an explanation of how the demand for lumber products was utilized to generate various lumber demand scenarios.The subsequent section details the procedure for executing the models and elucidates how the problem data were input to explore variations in the problem.

Problem setting
The OLPP was configured with 10 log diameters, a library of 100 sawing patterns, sawmilling and anti-stain treatment processes, and 10 anti-stain lumber.The OLPP was approached using a lumber planning methodology, which is comparable to a lumber scheduling approach.For lumber planning, a timeframe of six periods, corresponding to weekdays, was employed.In contrast, the lumber scheduling approach utilized a period of 138 hours, equivalent to 6 days of 23 hours each.Each order consisted of a set of 10 lumber products, and the volume of every order matched the demand for the lumber products that needed to be fulfilled within a specific period for the lumber planning problem.This volume equivalency was translated into an order with a due date specified in hours to be satisfied within the scheduling problem.

Lumber manufacturing data
The data set utilized in this study was sourced from a lumber manufacturing company in southern Chile.To uphold company confidentiality, certain aspects such as logs, labor, and inventory costs were adjusted, although they closely approximate actual figures.Log supply was assumed to be unlimited to prevent infeasibilities in the model due to log availability, and to maintain a focus on the impact of static heuristics rather than other factors (Singer andDonoso 2007, Maturana et al. 2010).
An anti-stain process is incorporated into the lumber production of Radiata pine due to high proportion of sapwood, aiming to prevent fungal activity.Hence, sawing and anti-stain processes were considered as operations in this research.
The initial demand for lumber products was defined as the average production that the library of sawing patterns could produce when utilizing the entire sawmill capacity.The demand was based on a sawmill with a capacity of consuming up to 111 m 3 of logs per hour and producing up to 62 m 3 per hour of lumber.These figures varied based on log distributions and the sawing pattern library.This assumption aligns with the current working conditions of the company and is consistent with the case study by Maturana et al. 2010.To assess the impact of static heuristic (sequences) on the OLPP, five lumber product demand scenarios were generated based on the initial lumber product demand.These scenarios focused on varying rates applied to the initial demand between periods and among lumber products, with thirty values generated for each scenario (Table 1).
Table 1: Scenarios of lumber products demand.
These five scenarios, each comprising 30 sets of lumber product demands organized by period or order, were fed into the models described in the next section, leading to a total of In the E static heuristic, orders are scheduled in ascending order of due dates.The S static heuristic involves sorting orders by increasing processing times.The L static heuristic schedules orders by decreasing processing times (Table 2).
In the PL model, the demand for lumber product must be fulfilled within a specific period of the planning horizon (i.e., days of the week).Although the formulation did not explicitly include due dates, demand was allocated to certain periods.Thus, the model implicitly ensured demand satisfaction within a period, equivalent to fulfilling an order containing a set of lumber product demands for a specific a due date (Equation 1 and Equation 2).  :Vol. of logs (m 3 ) of diameter i, log grade j, sawn with sawing pattern k, to satisfy order p.

Prod _ :
i sw Sawmilling productivity (h/m 3 ) when sawing log diameter i.Therefore, two models were formulated to address the same OLPP problem from different perspectives (Figure 1).Initially, the PL model met lumber product demand per period, implicitly ensuring the due dates of the set of lumber orders constituting that demand.In each period, PL generated lumber to fulfill demand until reaching production capacity.
Simultaneously, it determined the volume of raw materials and products for inventory.While this approach ensures timely deliveries to customers, it is costly due to its underutilization of mill capacity, disregarding setup costs and the number of production batches needed to satisfy orders (Wery et al. 2014).Conversely, the PS model was formulated, wherein lumber product demands are consolidated into orders with specified due dates processed in a sequence.Orders are applied in a sequence based on either processing times or due dates, as suggested by Yaghubian et al. 2001.The periods of the formulation were dropped off, then an "earliness" constraint was added that ensured that the processing time of order p, and processing times of preceding times orders must be less than or equal to the due date of order p (Equation 3).Accordingly, this constraint compelled orders to be processed without violating their due dates, however, due to model infeasibilities, the earliness constraint was relaxed, allowing overdue orders without penalties.
The main difference between PS and PL lies in the fact that PS operates on a one-week equivalent period, whereas PL involves six periods, corresponding to days of the week.
Additionally, lumber products demand by period in PL are associated with orders containing a set of lumber that must be satisfied by a specific due date.criteria.However, these delays were factored into determining key model performance metrics (e.g., backlog ratios, earliness, or lateness ratios).In the PL model, the total backlogs divided by the lumber product demands determined a value known as the backlog ratio for sawn lumber and anti-stained lumber (i.e., A_BO1(%), and A_BO2(%) in the model).A lower ratio indicates a better solution, signifying lower backlog volumes relative to demand.
In the PS model, metrics were derived from earliness and lateness of orders.Order earliness or lateness in hours was determined based on the time an order was produced in relation to its due date (i.e., A1p, A2p, B1p, and B2p in the model).The hours of earliness or lateness for each order relative to its due date were used to calculate the ratio of earliness or lateness (%) (i.e., Tfea1p, Tfea2p, Tfde1p, Tfde2p in the model).The ratio of earliness or lateness for each order, multiplied by the order's volume, determined the early or late volume of the order (i.e., Ad_1_volp, Ad_2_volp, Ba_1_volp, and Ba_2_volp in the model).The summation of early and late volumes per order determined the total late or early volumes (i.e., AV1, AV2, BV1, and BV2 in the model).Finally, the summation of the earliness or lateness ratio of orders determined the overall earliness or lateness ratio, equivalent to the backlog ratio of the PL model (i.e., A_AO1, A_AO2, A_BO1, and A_BO2 in the model).
The PL model allows for backlogs that need to be produced immediately in the subsequent planning period (refer to constraints 2 and 3 of model PL in Appendix A).In contrast, the PS model accommodates overdue orders, resulting in delays relative to the due dates (i.e., hours of delay).Technically, these delays were translated into volumes using sawing and anti-stain process productivity, representing equivalent backlogged volumes in the PL model (constraints 40, 41, 50, 51, 54, and 55 of model PS in Appendix B).In the case of PS model, overdue orders and the corresponding backlog volumes did not incur penalties (for detailed formulations of the models, refer to Appendix A and B).

Analysis of the lumber manufacturing costs
The results were evaluated with consideration of the modeling approach, the applied static heuristic, and the size of lumber orders.For large orders, the lumber manufacturing cost exhibit a change of no more than 0,1 %.In the case of small orders, the lumber cost did not vary beyond 1,7 %, and for mixed orders, the cost did not change by more than 0,5 %.Orders with high variation experienced a cost change of no more than 0,1 %, while orders with low variation saw a cost change of no more than 0,2 % (Table 3).
Table 3: Lumber manufacturing costs for the planning and scheduling approaches.

Analysis of due dates and backlog
The impact of the relaxation on overdue volumes was more pronounced than on lumber manufacturing costs.Larger averages of delayed orders (in hours) and consequently, backlogged volumes, were observed with the PS-S and PS-L in comparison to PS-E.
However, the average backlogged volumes for the PL approach were also noteworthy.For instance, the PS-E had an average of backlogged volumes equivalent to 0,5 % of the volumes of lumber demand ordered.In the case of PL, the average backlogged volumes represented 17 % of the volumes of lumber demand ordered.For PS-L, the average backlogged volumes were 41 % of the volume of orders ordered, and for PS-S, the average backlogged volumes were 44 % of the volume of orders ordered, irrespective of the order size (Table 3).Figure 2 shows the distribution of backlogged volumes across model scenarios.The PL and PS-E models yielded cost results that were remarkably similar across various order sizes.The PS-E sequence closely resembled how the multi-period PL model handled lumber product demand during the period when the demand was initiated.Consequently, both the PL and PS-E models exhibited very similar costs.However, due to the heuristic nature of the E static heuristic, the PL model consistently generated lower costs.These findings align with the results reported by Maturana et al. 2010, where an LP multi-period model and a heuristic scheduling planning tool based on due dates were compared.However, they reported larger differences, such as 52 %, in costs determined by the LP model and the heuristic approach for lumber demand scenarios.This disparity may be attributed to the use of more extensive and specific data perturbations, like increasing the demand of a low-value product by 10 %.In contrast, this research employed random changes within certain rates applied to all lumber demand data.Therefore, the differences reported in this study were smaller; for instance, the PS-E cost was only 1,7 % higher than the PL cost when processing small orders.
As highlighted by Gaudreault et al. 2011 for industrial settings, the inevitability of overdue orders is acknowledged.Consequently, the effectiveness of planning approaches can be evaluated through late deliveries.The results revealed that the average ratios of backlogged orders were 0,5% for the PS-E, 17 % for the PL, 41 % for the PS-L, and 44 % for PS-S approaches, irrespective of the order size.These values align with those reported by Gaudreault et al. 2011, who observed ratios of backlogged volume of orders ranging between 26 % and 50 % when addressing a similar problem.However, their solution involved utilizing a planning-scheduling MIP and an LPT heuristic model with coordination protocols over an agent-based platform.
The PS-E approach demonstrated intriguing capabilities of both models (i.e., PL model and PS model).It exhibited cost saving when planning orders with high variations in the volume of orders, achieving a 0,2 % lower cost than the PL for the same order sizes.However, the PS-E proved equally efficient as the PL when processing large and small variations orders as well (Table 4).Consequently, the PS-E displayed a greater ability to reduce due date delays and, therefore, overdue volumes compared to the PL.
Furthermore, the PS-L approach could be employed when planning small orders resulting in a cost 0,24 % lower than the average of the PL for the same order size.This potential utilization of PS-L could be extended to these order sizes.
The PL model demonstrated high efficiency in addressing the lumber manufacturing production planning problem as a general lot sizing and scheduling problem with cost minimization (Maness and Adams 1991, Maness and Norton 2002, Singer and Donoso 2007, Vergara et al. 2015, Broz et al. 2019).LP proved to be a highly effective solution method when adjusting order sizes and testing sequences for order processing.However, this research showed that this model formulation did not ensure the most cost-effective production plan or the lowest backlogged volumes under all circumstances.
While the percentage differences in costs for the model scenarios were small, the economic impact of the PS-E approach could be significant, given the operational scale of this type of facility.Furthermore, the PS-E approach demonstrated the capability to reduce the backlog ratio close to zero, compared to the PL backlog ratio of 17 %.No penalties were imposed to backlogs due to their subjective nature; however, when shortages occur, the demand is either lost or backlogged with substantial costs.In manufacturing, shortage costs are estimated based on lost revenue.Gupta and Starr 2014 suggested a cost ratio of 1:4 between holding inventory cost and backlog cost, while West 1989 assumed that a reasonable value for a backlogged unit should be the delivery price.Therefore, considering the chosen criterion for penalizing backlogs and keeping in mind its limitations, the PS-E and PL approaches were benchmarked.As a result, if the developed planning method is adopted using a conservative backlog cost, a sawmill producing 300000 m 3 per year would reduce backlogged orders by 51 thousand m 3 (equivalent to a 17 % backlog reduction).Assuming a holding lumber cost of 2 $/m 3 , the annual savings would amount to $408000.
The set-up cost was excluded from lumber processing costs to avoid the incorporation of binary variables for modeling the fixed cost associated with changing production plans, opting instead for an LP formulation.This decision was primarily made to facilitate problemsolving within a reasonable timeframe.The application of this relaxation approach has been observed in previous studies, such as Maturana et al. 2010 andYaghubian et al. 2001 in the context of lumber manufacturing production planning and dry kiln scheduling problems.
Additionally, the negligible setup times attributed to high-production sawmills, which are now under CNC control for log processing and cant breakdown machines, set-up times are negligible.

Conclusions
The PL approach resulted in the lowest costs, albeit with only a slight advantage compared to the PS-E.However, the PS-E approach demonstrated comparable efficiency to PL but with lower backlogged volumes.Consequently, the PS-E model is recommended as the best approach for planning lumber production orders in sawmilling operations regardless of lumber order sizes.It is important to note that this conclusion is applicable under specific conditions, including a cut-to-order lumber production environment, a significant emphasis on backlogs valuation or penalization due to overseas lumber customers' focus or specialty lumber products focus, and short-term planning horizons (e.g., Chilean sawmilling industry).The limitations of this research should be acknowledged.Firstly, only 30 model runs were conducted by order size, and an increased number of data sets and model runs would enhance the reliability of the results.Secondly, the data sets were generated based on the ideal lumber demand for this specific sawmill case study, and rates of variation were randomly selected.
Consequently, a more in-depth analysis of these rates of variation should be further studied.
Lastly, decision-making models for sawmilling and anti-stain operations were formulated, excluding kiln drying operations.Incorporating such operations could prove challenging due to their utilization of parallel resources.
. A simulation planning method simulated the arrival of new orders, demand, capacity, inventories, rolling planning horizon, and coordination mechanisms, resulting in an increased service level compared to centralized planning methods.Furthermore, Vergara et al. 2015 pointed out that the lumber planning problem has been tackled with conflicting objectives, where a solution optimized for one objective adversely affects another.To address this conflict, Broz et al. 2019 applied a Goal Programming formulation, balancing problem metrics to showcase the importance of reconciling decisionmakers' local objectives.Additionally, a model feature facilitating the planning of log transportation to the sawmill was introduced by Vanzetti et al. 2018.However, realistic modeling formulation in the cut-to-order environment discussed would have required the inclusion of slack variables and penalties.
used a MIP model, benchmarking results with an artificial intelligence heuristic procedure based on a Tabu search algorithm, demonstrating promising performance when compared with an exact problem solution.Meanwhile, Marier et al. 2014 formulated MIP models to address the OLPP.The MIP modelfor scheduling batches of lumber was simplified by imposing the LPT sequence to process orders on machines rather than generating and testing many combinations of order schedules.
Two planning and scheduling models, namely a multi-period planning model (PL) and a planning-scheduler model (PS) with 3 versions, were developed and performances were compared.The initial optimization model (PL) is a MIP model (PL) that was transformed into a single-period allocation of orders problem, referred to as planning-scheduler (PS).The PS includes three versions: a scheduling model with Earliest Due Date static heuristic (PS-E), a scheduling model with Longest Processing Time static heuristic (PS-L), and a scheduling model with Shortest Processing Time static heuristic (PS-S).
stain treatment production (m 3 ) of lumber product o in period s._:os Bat Backlog volume (m 3 ) of anti-stain treated lumber products o in period s.

Figure 1 :
Figure 1: Description of solution approaches for the lumber production problem

Figure 2 :
Figure 2: Lumber Manufacturing a) Costs Ratio and b) Backlog lateness ratio.
Integrated forest companies cater to customers with varying due date requirements, placing a significant emphasis on internal customer with high due date flexibility.Most of the lumber production planning research, however, has been conducted without accounting for any due date flexibility.Typically, only backlogs in terms of the volume of lumber products have been considered.The treatment of overdue orders may vary, contingent on the willingness of customers to accept delays.A promising avenue for new research would involve exploring approaches to model the problem with flexible due dates for customers, as opposed to excessively constraining the lumber manufacturing production planning problem.