Optimization of economic indicators in a carpentry... Revista Árvore 2021;45:e4519 1 OPTIMIZATION OF ECONOMIC INDICATORS IN A CARPENTRY COMPANY THROUGH INTEGER PROGRAMMING

This study aimed to analyze the economic feasibility of manufacturing frames in a carpentry company through two scenarios, including the production before and after modeling using operational research. The annual costs and revenues were quantifi ed considering a horizon of 30 years. The economic analysis included the following criteria: Net Present Value (NPV), Equivalent Periodic Value (EPV) and Revenues Cost Ratio (R/C). The modeling was applied after analysis of the standard production using an objective function to maximize the company’s profi t based on the quantity of each assortment to be produced. The standard production showed economic feasibility for all evaluated indicators, however implementing the operational research provided an increase of 57.54% for the NPV and EPV indicators, and 11.29% for the R/C ratio. In applying the sensitivity analysis after modeling, it was noticed that the value obtained for the NPV indicator was higher, even with a 10% increase in labor costs or in the price of wood, when compared to the standard production scenario before the modeling and without applying sensitivity analysis.


INTRODUCTION
Wood played a prominent role in the production of frames in past times. Metals only gained space in window and door manufacturing after the industrial revolution with the obtaining of steel on a large scale and at an aff ordable cost (Miotto, 2002). Thus, it is essential to rationalize the entire production process in order to ensure competitiveness for using wood in making frames.
Another important aspect is the fi nancial planning in activities related to wood processing through economic analysis, which can survey costs and revenues in order to obtain inferences about the viability indicators. Silva et al. (2005) reports the Net Present Value, Equivalent Periodic Value and the Revenue Cost Ratio among the most relevant economic indicators.
Given the countless possibilities of producing diff erent assortments in a carpentry company, such as: cladding; tilt and turn windows; windows/ window frames; frames/casings; doors/door frames; doors; skirting boards/baseboards, among others, the planner needs to be guided in relation to the best way to use their raw material (wood) in order to consolidate an optimal, consistent and sustainable profi t, and therefore must pay attention to the study of operational research. Thus, modeling should be taken as a parameter which can promote maximizing the company's profi t through an objective function and the respective restrictions based on the quantity of each assortment to be produced, thus guaranteeing better performance for the abovementioned viability indicators.
This study aimed to conduct an economic analysis on the wooden frame manufacturing process in a carpentry company, considering production scenarios before and after applying operational research.

Data collection
The study was carried out in a carpentry company located in the municipality of Vitória da Conquista, BA, Brazil, in the southwest of the state. Data collection took place through interviews and questionnaires considering the characterization of the raw material, the industrial process and the commercialization of assortments. Annual costs and revenues for the year 2020 were quantifi ed. The data were projected over a 30-year horizon due to the company's standard production fl ow, a period considered as the useful life estimation by carpentry.

Determination of annual net profi t
The annual net profi t was obtained via a cash fl ow, considering annual revenues and production costs. Costs were related to labor, wood and other inputs (glue, sandpaper, nails and energy), depreciation, machinery maintenance, fuel for product delivery and administrative costs, totaling US$63,141.35 annually ( Table 1). The raw material (wood) cost was calculated considering the product sum of the following variables: average volume of wood per assortment, numbers of each assortment made annually (Jequitiba and Jatoba wood) and the price of the cubic meter of these woods. The depreciation cost was determined using the straight-line method, taking into account the useful life of 30 years and a residual machinery value of 5.00% in relation to the acquisition price. Revenues were earned considering the quantity of each assortment off ered and their respective market price.
Currency units were converted into dollars in order to stabilize costs and revenues over the planning horizon. The following conversion was used (R$5.40 = US$1.00), the amount obtained according to the quotation made in February 2021.

Economic analysis
An economic feasibility analysis was performed with costs and revenues for the project under study using the following indicators: Net Present Value -NPV, diff erence between the present value of revenues the present value of costs (Silva and Fontes, 2005); Equivalent Periodic Value -EPV, which transforms the NPV into a fl ow of revenues or periodic and continuous costs equivalent to the current value during the life of the project; and the Revenues-Cost Ratio (R/C), defi ned as a ratio between the current value of the revenues and the current value of costs (Silva et al., 2005), according to equations 1, 2 and 3, respectively. Eq.1 Eq.2 Eq.3 In which: Rj = current value of revenues, Cj = current value of costs, i = interest rate, j = period in which revenues or costs occur, n = number of periods or duration of the project, t = number of capitalization periods between occurrence of two installments.

Operations research
Modeling was applied in order to maximize the carpentry company's profi t based on the quantity of each assortment produced (decision variables) using integer linear programming. The following technical restrictions were considered: the annual production capacity; a guarantee of at least 80% of the quantity already produced in the standard process (for the most demanding assortments) and the annual wood volume consumed by the company.
The company has a rustic production profi le, and the products are made according to the protocols established by the establishment itself. Production is standardized and of low capacity, mainly due to the reduced number of employees, which means that the company does not normally face demand restrictions for its products.
Due to the obstacles in the production process, we tried to limit the quantity of products made annually for the two species used considering the current production levels (technical restrictions 1 and 2); however, with prospects of changes in the quantities of each assortment produced by the business (doors, cladding, baseboards, tilt and turn windows, frames/casings, doors/doorframes and windows/window frames) in order to obtain a better return on investments and optimized scenarios for economic indicators. This was possible due to the diff erent prices of the products.
Considering the greater acceptability of some assortments in carpentry, restrictions were designed so that at least 80% of the quantity already produced in the standard process was ensured, including for windows/window frames; frames/casings; doors/door frames, and doors in these cases (technical restrictions 3 and 4).
Due to the restrictions on the productive capacity of the carpentry company, it was also sought to limit the wood volume consumed annually considering the current production levels for the two species used (technical restrictions 5 and 6).
The models were elaborated based on the two sources of raw materials (Jequitiba and Jatoba wood). Considering that the decision variables must necessarily assume integer values (quantity of each assortment to be produced), the @gin function was used in the Integer programming (IP). Although rounding solutions can often seem the easiest and most practical solution to large-scale forest planning problems, this practice can lead to unsatisfactory results and often to unviable solutions, thus justifying the use of other more specifi c methods such as integer programming (Silva et al., 2003). After building the model, the Lingo version 3.0 program was used to obtain the data.

Production costs
The respective market prices of Jequitiba and Jatoba woods of US$277.77/m³ and US$1,055.55/m³ provided a total annual cost of raw material (wood) to the order of US$24,822.21, corresponding to 39.31% of the annual company budget. This fi gure was the second most representative. The labor cost represented 51.73% of the annual expenses (Table  1). It was found that machinery depreciation was the least impacting activity in the composition of costs, representing 0.39% of the total (Table 1). The thicknesser, planer and tenoner were the equipment Table 1 -Annual cost in the frames manufacturing process in the evaluated carpentry company. Tabela 1 -Custo anual do processo fabril de esquadrias em madeira na carpintaria avaliada. which most contributed to this cost composition due to the higher acquisition values, representing 71.57% of the total.

Revenues
A total of 1,008 and 564 products were produced from Jequitiba and Jatoba wood, respectively. The amount of revenue obtained annually for the two species was US$47,863.08 (Jequitiba) and US$30,654.60 (Jatoba). Frame/casing production was responsible for the greatest impact on revenues, with 39.00% and 91.34%, respectively for the Jequitiba and Jatoba woods.

Application of the operations research
After modeling and obtaining the values for the decision variables, namely the quantity of each assortment to be produced to maximize profi t (Table  2), a comparative table was elaborated in relation to the production process before and after applying the operations research, considering the Jequitiba and Jatoba woods as raw materials (Table 3).
The algebraic formulas of the modeling are shown below. Eq.4 In which: X i : the decision variables referring to the quantities of each assortment produced annually (i), with these being: X 1 : Qty. of claddings; X 2 : Qty. of tilt and turn windows; X 3 : Qty. of windows/window frames; X 4 : Qty. of frames/casings; X 5 : Qty. of doors/ door frames; X 6 : Qty. of doors; X 7 : Qty. of skirting boards/baseboards. Jequitiba wood (including all assortments), according to the production capacity of the carpentry company.
Smax[2]: Technical restriction 2 regarding the maximum quantity of products made annually with Jatoba wood (including all assortments), according to the production capacity of the carpentry company. "0.8 Xi" : technical restriction 3 (guarantee of at least 80% of the quantity already produced in the standard process for the most demanding assortments) for Jequitiba wood, with these being: window/window frames (x 3 ); frames/casings (x 4 ); doors/door frames (x 5 ) and doors (x 6 ). "0.8 Xi" : technical restriction 4 (guarantee of at least 80% of the quantity already produced in the standard process for the most demanded assortment) for Jatoba wood, in this case only including frames/ casings (x 4 ).

Eq.6
In which: Vmax[1] : Technical restriction 5 referring to the maximum volume of Jequitiba wood consumed annually according to the carpentry company's production capacity.

Vmax[2]
: Technical restriction 6 regarding the maximum volume of Jatoba wood consumed annually according to the carpentry company's production capacity.
Xi: Assortment quantity produced annually (i); V i = Unit volume of the assortment (i);

Economic analysis
The results showed attractive for all evaluated indicators when analyzing the standard production fl ow, however implementing the operations research provided increases of 57.54% for the NPV and EPV indicators, and 11.29% for the R/C. The EPV in the standard process showed that the project has the capacity to generate annual revenues to the order of US$15,515.32; however, this revenue was presented in the operations research to the order of US$24,443.96 (Table 4).
In analyzing operational research through integer programming when converting the forest into multiproduct, Campos et al. (2013) found an average Table 2 -Quantity of each assortment produced by carpentry company from Jequitiba wood (A) and Jatoba wood (B) after application of the operational research. Tabela 2 -Quantidade de cada sortimento a ser produzido na carpintaria com madeira de Jequitibá (A) e Jatobá (B), após aplicação da pesquisa operacional.

Sensitivity analysis
An additional 2% and 4% were considered in relation to the interest rate adopted (6.00%), as well as fl uctuations of -10%, -5%, + 5% and + 10% in labor and wood costs. The sensitivity analysis was designed for both scenarios (before and after modeling), as noted in Table 5.
Additions of 2% and 4% in the adopted interest rate resulted in NPV reductions of US$38,897.65 In relation to wood cost fl uctuations, a 10% reduction provided an addition to the NPV to the order of US$34,151.92. Considering a 10% drop in labor costs, this addition was even greater for the respective indicator (US$44,957.32).
In analyzing the sensitivity projection after applying the operations research, it was observed that even in the most critical scenarios (10% interest rate; 10% increase in the cost of wood or labor), the NPV, EPV and R/C indicators were still superior when compared to standard values without sensitivity analysis and before implementing the operations research (Table 5).

Revenue in the production process
The cladding, tilt and turn windows and baseboards proved to be less expressive in the composition of total revenue, mainly due to low demand and lower market prices in relation to the other assortments (Table 3). The lower market value for Jequitiba wood provided greater attractiveness for the assortments from this species (Tables 3).

Application of the operations research
Mathematical modeling based on Integer Linear Programming needs to be further explored in the forestry area, and above all in works regarding wood processing, aiming to optimize and maximize revenues.
Considering in the modeling in this study a production of at least 80% of the quantity already produced in the standard process for the most demanding assortments for Jequitiba wood (windows/ window frames; frames/casings; doors/door frames and doors), a maximum amount of products made annually (1,008), for a volume of 35.40 cubic meters according to one of the working conditions of the evaluated carpentry company (Table 3), the company would have to stop producing cladding in order to maximize profi t.
Cladding presented a low market value and economically did not add advantages in optimizing revenues in relation to other products made by the company; therefore, its production is not prioritized in the post-modeling scenario (Table 2).
Considering in the modeling in this study, it would still need to increase the quantity of window/ window frames by 218.75% and reduce the frames/ casings, door/door frame, and door production by 20% (Tables 2 and 3).
The demand for the assortments made in the carpentry company proved to be very active because it is a small quantity of products (1,008 and 564 specimens per year for jequitiba and jatoba wood, respectively) ( Table 3). The change in the production scenario proposed by the operations research to optimize the economic indicators would not compromise the demand of the evaluated carpentry company, because *WSA: Values Without Sensitivity Analysis (interest rate 6% p.a, annual labor cost US$32,661.00 and annual wood cost US$24,822.21). *SAS:Valores Sem Análise de Sensibilidade (taxa de juros 6% a.a, custo anual de mão de obra US$ 32.661,00 e custo anual de madeira US$ 24.822,21). in addition to the low supply capacity, the regional insertion of the company (logistics) contributed to high demand for the assortments produced.
Also considering a production of at least 80% of the quantity already produced in the standard process for the greater demand assortment for jatoba wood (frames/casings), maximum quantity of products made annually (564), with a limit volume of 14.20 cubic meters of annual wood according to one of the working conditions (Table 3), the company, aiming at maximizing profi t, would have to stop producing baseboards due to the low market value and reduce the number of frames/casings by 108 units and increase the number of doors/door frames by 35 units (Tables  2 and 3).
Such assortments prioritized in modeling would not be subject to demand restrictions, mainly due to the small supply potential of the evaluated company.

Economic analysis
Rationalizing and optimizing in the forestry sector can be consolidated as an alternative to reduce production costs. In analyzing the loss process in sawmill, Vital (2008) observed a maximum yield of 55% for hardwood, thus evidencing the importance of optimizing resources to reduce costs.
Effi ciency in the process of transforming raw materials into products by the wood sector is fundamental for reducing production costs (Biasi and Rocha, 2007) and optimizing profi ts. One of the factors which contribute to economic unfeasibility in the wood processing segments is related to the high rates of losses. Wood generates waste when it is processed and transformed, culminating in a considerable waste of raw material, which does not add value to a material considered so noble (Garcia et al., 2012). In this regard, maximizing economic indicators in carpentry industry promoted by operations research through changing the production scenario attenuates loss and waste processes, promoting greater economic sustainability for this sector.
The labor and wood costs represented a signifi cant potential for annual production costs of the carpentry company (Table 1), being considered strategic in economic planning of the enterprise.
It was noticed that the change in the production scenario with the modeling application (maintaining the same quantity of manufactured product) provided a considerable increase in the NPV, EPV and R/C values. This context demonstrates that it is possible to enhance the company's revenues without the need to expand its production process, simply by directing the production focus through operations research techniques. Jequitiba wood presented a more promising response in terms of additional revenue when compared to Jatoba. This was due to the fact that this species contemplates a greater diversity of assortments, which allowed for better effi ciency in reorganizing the production scenario, providing more effi cient gains with modeling (Table 3).

Sensitivity analysis
The sensitivity analysis aimed to contextualize possible scenarios which may constitute a real situation for the evaluated enterprise, being of paramount importance for planning. When the sensitivity analysis was applied to production optimized by the operations research, it was noticed that even with an increase in the annual interest rate from 6% to 10%, NPV was higher when compared to that obtained in the standard scenario (before modeling) and with an interest rate of 6% per year (Table 5). This indicates that the operations research was not only able to promote more consistent gains, but also to resist possible fl uctuations in interest rates.
The economic indicators showed signifi cant sensitivity in relation to the market price of wood. This can be explained by the fact that the carpentry company works with noble species of high commercial value. The evaluated sensitivity scenarios did not promote unfeasibility in the project (interest rate, market price of wood and labor cost), even in the standard condition before applying the operations research, which can prove the economic sustainability of the carpentry company.

CONCLUSION
The carpentry proved to be viable for all the analyzed criteria, including the sensitivity analysis. Operational research can be applied to the economic planning as a way to promote sustainability for this sector. Despite the lower market value for assortments from jequitiba wood, this raw material proved to be strategic in the composition of the assessed carpentry revenues, mainly due to the greater competitiveness in relation to the market price of jatoba wood. The regional insertion of the carpentry company in a highly structured commercial hub, coupled with its reduced supply capacity, contributed to the fact that the evaluated undertaking did not suff er major restrictions on the demand for its products.