Linear Programming Approach for Optimal Profit Mix of Cassava Grates and Flour

Article history: The research aimed to identify the optimal combination of cassava grates and flour to maximize the profit of a project initiated by the Philippine Root Crops Research and Training Center at Visayas State University. The findings revealed that the most profitable combination consisted of 125 kg of cassava grates and 25 kg of flour, resulting in an optimum profit of Php 4,250.00 for the entire production. The study highlighted that the project was constrained by limited inputs, preventing it from exceeding the optimal quantities of the two products. To enhance the profitability of both products, adjustments in the resources utilized during production were necessary. While the current production of cassava grates falls below the optimum quantity, indicating room for increased production with the existing inputs, the production of cassava flour had already reached its optimal level. To achieve optimal quantity and profit, the project must implement strategies to prevent resource wastage.


INTRODUCTION
Cassava (Manihot esculenta Crantz) or locally known as "kamoteng kahoy" in the Philippines is an important perpetual root crop that provides food for over 500 million people in the world [1] [2].It is grown primarily for its starchy tuberous roots and is considered one of the essential tuber crops, serving as a dietary staple food for various populations in the tropics [3].Cassava has undergone various processing methods, resulting in the production of starch, flour, traditional dried cassava, chips, and diverse non-food items like adhesive for charcoal briquettes, animal feed, paper, and textile [4].Cassava has a strong adaptive advantage compared to rice since it grows where soils are marginal in fertility and rainfall is uncertain [5].These reasons have made cassava into a crop of primary importance for the food security of farmers living in fragile ecosystems and socially unstable environments [6] [7].Despite the cassava's high demand as a staple food, its high perishability is one its disadvantages [8].Hence, the Philippine Root Crops Research and Training Center (PRCRTC) have developed cassava grates and flour as an answer to the crop's high perishability. https://doi.org/10.53893/fms.v1i1.257 The attainment of the optimal profit mix for cassava grates and cassava flour holds significance as it ensures effective resource allocation and maximizes the overall profitability of the production process.Striking an appropriate balance between these two products is essential for meeting market demand, utilizing available resources efficiently, and enhancing economic viability.The realization of this optimal mix is instrumental in fostering sustainable cassava processing, facilitating income generation, and contributing to food security.
Linear programming is necessary to determine the optimal combination of processes needed to produce the desired level of output.Linear programming model is generally used to achieve optimum solutions under several limitations by studying maximum or minimum objective functions for any given situation [9] [10], including profit maximization in an industry [11] [12].Thus, it is on this premise that linear programming approach was used to determine the optimum quantity combination of producing cassava grates and flour to maximize profit.

Approaches to Solving Linear Programming Problem
Linear programming problems were addressed through graphical techniques and SIMPLEX algorithms.The graphical method involved solving a function with two variables in linear programming, where each inequality constraint was represented graphically as an equality constraint.Figure 1 depicted the linear presentation of constraints alongside the highest profit line (objective function).

Legend:
A= constraint 1 B= highest profit line C= constraint 2 D= optimum combination The Simplex method was regarded as one of the fundamental techniques, serving as a foundation for numerous linear programming methods, either directly or indirectly derived.It operated through an iterative, stepwise process, gradually moving towards an optimal solution aimed at maximizing or minimizing the objective function.As depicted in the figure, both the constraints and the objective function underwent sensitivity analysis, recognizing that these elements are not fixed and may undergo changes that can impact the parameters of a linear programming (LP) model.The diagram showcased the computation of various components in formulating the LP model, employing graphical methods, the simplex method, and the LINDO software.

Conceptual Framework
The study specifically focused on finding optimal solutions for two-variable problems, making the graphical method under linear programming applicable.Additionally, the simplex method, a general-purpose linear programming algorithm commonly used for solving large-scale problems, remained relevant to this study [13].

Respondents and Site Selection
This study engaged informants comprising staff, laboratory personnel, and laborers actively engaged in the production of cassava grates and flour.These informants possessed firsthand knowledge of the production processes, operational flow, and production levels at specific intervals.The research took place at the Philippine Root Crops Research and Training Center, situated within the Visayas State University campus in Visca, Baybay City, Leyte, as the production site was located within this institutional setting.

Gaining Entry to the Research Site
To facilitate the execution of the research study, the researchers sought permission from the staff overseeing cassava grates and flour production at the Philippine Root Crops Research and Training Center.A formal certificate of permission, signed by both the research adviser and the department head of Business and Management, was obtained for this purpose

Data Gathered
This research employed a combination of primary and secondary data sources.The primary data encompassed various time periods associated with the production of cassava grates and flour, including the grating period (60 minutes), spinning period (120 minutes), pulverizing period (45 minutes), drying period (720 minutes), milling period (150 minutes), production period (960 minutes), labor time (2640 minutes), and raw material (Cassava) usage at 400 kg per production cycle.Additionally, key financial elements such as markup price or contribution margin, selling price, variable cost per unit, and total production cost were considered.The study also factored in demand for the products within the locality.To derive the contribution margin as the coefficient for the objective function, the research utilized a series of equations involving these parameters.This comprehensive approach allowed for a thorough analysis of the production process and financial aspects associated with cassava grates and flour.

Data Gathering Procedure
In a systematic manner, data collection for this research involved interviews with laboratory personnel.Comprehensive field notes were diligently recorded while observing the entire production operations at the research site.Additionally, supplementary information was obtained through the review of relevant literature and online resources accessed through the library and the internet.

Data Analysis
Linear programming approach was employed in this research.The data gathered were analyzed through graphical and simplex method with the aid of Linear Interactive Discrete Optimizer (LINDO) software [14].
The decision variables, denoting the amounts of inputs or outputs within the decision maker's control, are termed as decision variables.In this research, these decision variables were denoted by X 1 and X 2 , representing the quantities in kilograms of the two respective products.Simultaneously, the objective function, expressed as Z=NX 1 +NX 2 , served as the performance indicator for optimizing the production of the two products.The objective function underwent scrutiny within the framework of specific constraints, delineating limitations on the overall production of cassava grates and flour: A Linear Programming Model is composed of three essential elements: the decision variables, the objective function and the constraints.The elements interlinked and must be considered together linearly (Manish).

Philippine Root Crop Research and Training Center profile
[27] The Philippine Root Crop Research and Training Center serves as a governmental institution dedicated to research, development, and training related to various root crops, such as cassava, sweet potato, taro, yam, yambean, arrowroot, and others.Established on March 21, 1977, by Presidential Decree 1107, PhilRootcrops is situated at the Visayas State University (VSU) in Visca, Baybay City, which serves as the zonal agricultural university of Visayas.As a pivotal entity, PhilRootcrops holds the leadership role in the National Rootcrop RDE Network-a collective of institutions tasked with planning, implementing, coordinating, monitoring, and evaluating research and development/extension programs to support the rootcrop industry.

Process Flow of Producing Cassava Grates and Flour
The production of cassava grates and flour involved a series of essential processing steps, as illustrated in Figure 3.The diagram emphasized the utilization of fresh cassava roots to ensure product quality and prevent issues like cassava stricking, which could lead to toxin production.Primary processing of the fresh cassava roots included sorting, peeling, and washing before chipping or grating.Following grating, the subsequent steps involved spinning, pulverizing, and drying, aiming to eliminate extract or water content for efficient drying.Drying could be achieved through sun-drying or mechanical dryers [15]; historically, cassava chips for flour were sun-dried, while mechanical drying was used for grates.However, recent practices favored mechanical drying for both products due to its time efficiency, particularly when weather conditions were unfavorable for sun-drying, thereby preventing disruptions in cassava flour production [16].
In the case of cassava grates, the process was relatively shorter, involving packing immediately after drying to produce the final product of fine grates.On the other hand, cassava flour underwent an additional step post-drying [17].A recent modification in the production site involved making cassava flour directly from the cassava grates produced earlier in the process, as opposed to the previous approach where cassava flour had a separate production process.This adjustment was driven by the demand-driven nature of cassava flour production.The milling process followed the production of cassava grates, leading to the final stage of packing the finely produced flour.

The Objective Function
A linear programming algorithm required a singular objective, which could involve either maximization or minimization [18].This objective served as a mathematical expression used to calculate the total profit or cost for a given solution.In the context of this study, the objective function was formulated to maximize the profit in the production of cassava grates and flour.

ISSN: XXXX-XXXX
The objective function, Z=28X 1 +30X 2 , was derived to maximize profit by determining the contribution margin per unit.This margin was obtained by subtracting the variable cost per unit from the selling price.The unit contribution margin was employed as a coefficient in the objective function to ascertain the maximum profit in production, given that the contribution margin was equivalent to the markup price.Markup price involved adding a constant percentage to the cost price of an item to determine its selling price.Hence, the formulated objective function, Z=28X 1 +30X 2 , measured the overall performance of the production of the products.

The Constraint
Constraints referred to limitations within the equation's process that imposed restrictions on the potential values that decision variables might assume [19].Two distinct types of constraints were identified: structural and non-negativity.[28] Structural constraints encompassed the resources involved in production, including grating, spinning, pulverizing, drying, milling, production period, labor time, and the raw material, which was cassava.Meanwhile, non-negativity constraints imposed limitations ensuring that variables possessed values greater than or equal to zero in Table 2.

Optimum Profit
The contribution margin for each product was applied to the optimal combination of grates and flour to compute the maximum profit for both products.The table (Table 3) showcased the optimal quantities and profits derived from the production of cassava grates and flour.The presented table indicated that to achieve the optimal overall profit of PhP 4,250.00 in the entire production, the project should have manufactured 125 kg of cassava grates and 25 kg of cassava flour.

Graphical Presentation of the Optimal Profit
The graphical presentation below illustrated the optimal combination of the two products leading to maximum profit.Within the feasible solution space, where all constraints were met, the optimal solution occurred at the intersection of the milling boundary and the line formed by the binding constraints (grating, spinning, pulverizing, and drying period, labor time, and raw material).The point within this feasible solution space, furthest from the origin, precisely determined the optimal combination of the two products for maximizing profit.
Thus, the algebraic determination of the optimal combination involved eliminating one of the equations representing the binding constraint and the milling constraint.This process resulted in a combination of 125 kilograms for X 1 , representing cassava grates, and 25 kilograms for X 2 , representing cassava flour.Substituting this combination into the objective function yielded the optimal profit of Php 4,250.00.

Reduced Cost
The term "reduce cost" signified the rate at which the optimal objective function value changed with a one-unit increase in a non-basic variable [20].In the context of this study, both X 1 and X 2 exhibited a reduced cost of zero (0).

Constraints Slack and Surplus Situation
Slack occurred when the left side of a constraint was less than its right side, indicating the amount needed to fulfill the available resources in production [21].Conversely, surplus arose when the left side of the constraint exceeded its right side, indicating an excess beyond the requirement.
The study encompassed eight constraints governing the production of cassava grates and flour.The first seven constraints regulated processing in terms of the duration each machine operated, the production period, and the time laborers dedicated to a specific production cycle.The eighth constraint related to the quantity restricting the production of cassava grates and flour.
In this study, the constraints involving grating, spinning, pulverizing, drying, and milling periods, labor time, and raw material (cassava) were regarded as binding constraints, signifying that both slack and surplus were equal to zero.This suggested that the available resources outlined in the constraints had been fully utilized.However, the production period constraint indicated a slack situation with a value of 120, indicating that the available production time was not fully utilized (Table 4).

Dual Price
The shadow price referred to the extent by which the objective function value would alter with a one-unit change in the right-hand side (RHS) value of a constraint [22].This phenomenon was observed exclusively in binding constraints.In this study, the dual values, or shadow prices, of the milling period and labor time constraints were determined to be 3.444444 and 1.414141, respectively, indicating that these ISSN: XXXX-XXXX constraints were binding.Furthermore, any modification to the RHS value of these binding constraintsspecifically, milling period and labor time-would result in a corresponding change in the optimal value of the objective function (Table 5).
The milling period constraint exhibited a shadow price of 3.444444, suggesting that an additional profit of 3.444444 could be achieved by extending the milling period by one minute.The same principle applied to the binding constraint of labor time.
In essence, the impact of changes in the RHS value of a constraint on the optimal value of the objective function could be determined by multiplying the amount of change by the constraint's shadow price [23] [24].It was crucial to note that this held true within a restricted range of feasibility.Within this range, both the shadow price and the optimal combination in the objective function remained constant, as long as the alterations made to the RHS values of the milling period and labor time constraints fell within their respective feasibility ranges.

Range of Optimality
In the concept of linear programming, the notion of the range of optimality was explored.It delineated the scope in which an objective function coefficient could fluctuate while preserving the existing optimal solution [25].A modification in the value of an objective function had the potential to induce a shift in the optimal solution to a problem.In a graphical solution, this shift would involve transitioning to another corner point within the feasible solution space.However, it's crucial to recognize that not every alteration in the value of an objective function coefficient necessarily led to a modified solution.Typically, there exists a range of values within which the optimal values of decision variables remain unchanged.In this specific case, the range of optimality for the coefficient related to cassava grates was limited to a decrease of 28 and an increase of 62.
Consequently, as long as the coefficient fell within this range, the optimal values persisted at 125 kg of grates and 25 kg of flour.Conversely, if the change extended beyond this range of optimality, the solution was no longer optimal.Regarding flour, the range of optimality allowed for an increase in the markup price up to infinity and permitted a decrease only up to 20.66.As long as changes stayed within this range, the optimal values of 125 kg of grates and 25 kg of flour were maintained (Table 6).

Range of Feasibility of the Right-Hand-Side values
The right-hand side (RHS) of a constraint typically denoted a constant value representing the maximum (<, =) or minimum (>) requirement.The allowable minimum/maximum RHS comprised sets of values for the RHS of a resource where the shadow prices remained constant.This range was also referred to as the range of feasibility since, within this interval of values, the solution to the current set of binding constraint equations was feasible.
When scrutinizing the RHS of a constraint, it was essential to ascertain whether a specific constraint was binding.In the production of cassava grates and flour, eight constraints were considered (grating, spinning, pulverizing, drying, milling and production period, labor time, and cassava).Among these constraints, grating, spinning, pulverizing, drying periods, labor time, and raw material (cassava) were deemed as binding constraints.These binding constraints exerted a limiting influence on the values of the objective function, essentially serving as restrictive conditions that may have impacted the optimal value for the objective function.
The range of feasibility was established by examining the allowable minimum and maximum values of the constraints.The constraints related to grating, spinning, pulverizing, drying, and production periods, as well as raw material (cassava), exhibited a feasibility range from 0.000000 to infinity.This signified that optimality persisted even if the values increased to any level, provided there was no decrease.The labor time constraint had a feasibility range from 2475.000000 to -0.000039.This implied that the labor time constraint

Figure 2 .
Figure 2. A Conceptual Framework on Formulating a Linear Programming Model subjected to Sensitivity Analysis The diagram presented in Figure 2 served as a valuable tool for determining the optimal combination of cassava grates and flour to maximize the profit of the Philippine Root Crops Research and Training Center.As depicted in the figure, both the constraints and the objective function underwent sensitivity analysis, recognizing that these elements are not fixed and may undergo changes that can impact the parameters of a linear programming (LP) model.The diagram showcased the computation of various components in formulating the LP model, employing graphical methods, the simplex method, and the LINDO software.The study specifically focused on finding optimal solutions for two-variable problems, making the graphical method under linear programming applicable.Additionally, the simplex method, a general-purpose linear programming algorithm commonly used for solving large-scale problems, remained relevant to this study[13].

Table 2 .
Constraints of the production

Table 3 .
Optimum quantities and profit of cassava grates and flour

Table 4 .
Slack and Surplus of Constraints

Table 5 .
Dual Prices of Constraints

Table 6 .
Optimal Range of Objective Coefficient