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INTRODUCTION
"No dreams come true until you wake up and go to work" (Amish,n.d.). Most business or planning problems involving resources can be converted into mathematical problems. Similarly, any problem can only be resolved if we fix and try to solve it. All businesses strive to find the most effective way to operate in order to make the most profit with their limited resources. Therefore, optimal production planning and care are required to sustain such an organization's optimal profit-making and existence. Production planning involves implementing various activities and measures to ensure optimal production that satisfies customer demands because the natural world has limited resources. Profit-making is the goal of every industry, company, firm, and even small business, as that will guarantee its existence. Companies aim to bring big profits to their advantage for their continuous existence, productivity, and expansion. The key to making profits in manufacturing industries lies in producing goods at minimum cost and maximum profit that is the correct standard quantity and at the right time, especially for sustainability and growth (Oladejo et al., 2019). Large corporations may dominate the stock market. However, the economy is kept afloat by start-ups and small enterprises, or what economists refer to as the primary sources of growth. Small firms present a challenge to formerly stale industries with their creative ideas and goods. Small businesses are more diverse in shape, function, culture, and potential because they are more adaptable and can be launched by almost anyone with grit and a creative concept. Moreover, because they operate locally, they create job opportunities for locals/neighbors, which could bring profit to the locality. Many
Filipinos give up their jobs mainly because they want to build their businesses. Several Filipinos want to be their boss and control their time. Owning a business provides advantages such as earning more than usual in the standard 9-5 jobs. The Philippines is considered a developing country in the Asian continent. It has rich natural resources, raw materials, and especially the workforce. Its present growth can be attributed to the resilience of the Filipino spirit and the industry of millions of overseas workers. A business venture can be exciting and challenging if it is in an area with a strong cultural heritage and people wishing to overcome poverty. The economy heavily depends on the remittances of overseas Filipino workers (OFWs) and exports of goods like sugar, pineapple, coconuts, and electronics products to developed nations like the United States, Europe, and other Asian nations. However, historians would probably refer to the last couple of years as the "lost period" not only for the country's economy but for the individual lives that have been turned topsy-turvy by the Covid-19 restrictions and enforced homestay, which left Philippine society in disarray. The global health crisis has dramatically impacted the economy, apparent in the slowdown of tourism, airline, hospitality, and even the retail industry. Supply chain trade has been disrupted. According to the Department of Labor and Employment, approximately three million Filipino workers lost their jobs due to the pandemic crisis' persistent epidemic as some enterprises came to a halt. This pandemic has brought about changes that are far beyond our comprehension. However, one basic fact remains: whatever dislocations to our lives, relationships, and institutions this crisis has wreaked, it can surmount these difficulties. That is why different small businesses appeared.
Importance of the Study This study aimed to provide crucial information and knowledge regarding profit maximization through Linear Programming techniques in running any business. Significance of the Study The result of this study is beneficial to the following: Students - This study will benefit the students in a way that will help them gain knowledge about Linear Programing that can be used to maximize profit. Business owner - This study would benefit business owners by knowing the importance of Linear Programming techniques to improve their decision quality by calculating the cost and the profit of different things. Moreover, to gain insight into maximizing the profit while minimizing the cost. Future Business Owner- This study will significantly benefit business owners to expand their knowledge and awareness about using Linear Programming to maximize their profit and develop their business skills. It will help them to be ready for this future undertaking. Future researchers - Future researchers will remarkably benefit from this study, which will serve as their reference and guide in conducting another related study. Through this study, they will be able to gain insight into how to maximize profit using linear programming. Objectives of the Study
This study aims to maximize Banana Cake's and Ube Cake's weekly profit on Nasnip's Pastry. Specifically, it will determine the following.
- To determine the weekly cost per product, the demand per product, and the time requirement to produce each product.
- To establish a model based on the data gathered.
- To find the optimal production plan based on the model produced. Time and Place of the Study This study was conducted in Purok 1 barangay Alawihao Block 18 lots 14 & 15, Villa Esperanza Subdivision Alawihao from 2022 to 2023 of Daet, Camarines Norte. Scope and Limitation This study only focuses on the profit maximization of Nasnip's pastry at Barangay Alwihao Daet, Camarines Norte. Banana cake and Ube cake are the main variables of the study. The model constructed was based on the data received by the researchers. The model of this study may not apply to other pastry businesses because of its limited ability to be generalized due to the uniqueness of the data used in its formulation.
offering services intending to sell them for profit. Businesses supply consumers with various products and services, including healthcare, automobiles, and countless others. Goods are material objects that companies produce, like laptops. Services are intangible products that can neither be kept nor stored by customers. Doctors, lawyers, hairdressers, car washes, and airlines offer such services. Businesses also provide equipment, products for resale, computers, and dozens of other items to hospitals, stores, governments, and other organizations. The primary motivation behind establishing a firm is profit. If a company does not make a profit, that could also mean that it has not accomplished its goals. Profit is capital for businesses to do many things. (Omokayode, 2020) stated that Earning profit is an essential aspect of business that helps the firm manage a secure financial position regardless of any risk. The organization generates the return from expenses and efforts made in business activities. Maximizing returns is essential for companies. After all, it helps the business manage successful operations with changing market trends because it enhances the ability to invest. Beyond profit, the business must improve and manage goodwill to maintain a corporate reputation and ensure customer satisfaction. Benefiting society is another aspect that needs to be a focus because it denotes a positive relationship of the firm with its key stakeholders, that is, customers. On the other hand, return is a critical concert because it denotes an organization's ability to operate effectively in changing business environment. Khurma (2022) describes linear programming, also abbreviated as LP, as a simple method used to depict complicated real-world relationships using a linear function. Linear Programming is a method of solving problems that involve a quantity to
be optimized (maximized or minimized) when that quantity is subject to certain restrictions. It is considered the most prominent operations research (OR) technique designed for models with objectives and constraints that are all linear functions. The variables in the mathematical model have a linear relationship. An LP model consists of three essential components: decision variables, objective function, and constraints. Relationships in the real world can be highly complex but can be represented using linear programming, making analyzing them easier. It can be characterized as an approach for the best-case scenario optimization of a linear function. Linear equality and inequality requirements make up this linear function or objective function. By minimizing or maximizing the goal function, we achieve the best result. Miller (2020) claims that linear programming effectively models various real-life problems, from routing airlines to transporting oil from refineries to towns, by finding the minimum cost to achieve the minimum standard required. In addition, there are many techniques for improving decision-making, such as optimization, neural networks, and queuing theory. Adebiyi et al. (2014) focused on linear programming for achieving product-mix optimization and optimum firm performance. Their result showed that only two of the five items they considered in their computational experiment were profitable. Ibitoye et al. (2015) empirically examined the impact of linear programming in the entrepreneur's decision-making process as an optimization technique for maximizing profit with the available resources. Zakariyya et al. (2022) used the linear programming method to examine the unit cost of production, the selling price, the quantity of various raw
minimizes a linear objective function that satisfies a set of linear constraints (linear equations and/or inequalities). Tsolas et al. (2018) propose a graphical method to identify and eliminate redundant generation and consumption of water and energy resources within a nexus. Using a graphical method, Tsolas et al. obtain the essential nexus configurations for a minimum generation or grid supply maximization. It was clear from the aforementioned related literature and studies that Linear Programming is applicable and helpful in optimizing profit in different kinds of businesses. The graphical method is the most basic and prominent technique applicable in various fields; it is an efficient method, which is why a more comprehensive is used in different sectors to get a possible result. This thought was supported by various research and strengthened by the research about linear programming. Theoretical Framework Multiple algorithms have been developed to solve linear programming models because of their importance in various sectors. The simplex method is one of the most famous methods in linear programming. This study will focus on the graphical method, one of the simplest methods for solving a linear programming problem (Bali, 2023). Linear Programming uses a mathematical model to describe the problem of concern. Thus, linear programming involves planning activities to obtain a result that reaches the specified goal best among all feasible alternatives. (Akphan & Iwok, 2016). According to Murray (2006), “An optimal as well as a feasible solution to an LP problem is obtained by choosing one set of values from several possible values of decision variables x1, x2,..
., xn, that satisfies the given constraints simultaneously and also provides an optimal (maximum or minimum) value of the given objective function.” Linear Programming Model The standard form of a linear programming problem has the following properties. i. All the constraints should be expressed as equations by adding slack or surplus variables. ii. The right-hand side of each constraint should be non-negative (if not). This is done by multiplying both sides of the resulting constraints by -1. iii. The objective function should be a maximization type. For n decision and m constraints, the standard form of the linear programming model can be formulated as follows: Methods for Solving Linear Programming Models The linear programming problem (LPP) can be solved using different methods such as:
METHODOLOGY
This chapter presented the methods and procedures used throughout the study. It included the research design, source of information, data-gathering tool, and procedure. Research Design In order to get the information from the respondent, the researchers used the descriptive analytical method. This method is helpful since it focuses on discovering ideas and insights and summarizes past data to identify patterns or meanings. This design helped the researchers achieve the desired data and the information needed. The primary data were gathered using an interview guide, and the data serves as a basis for conducting the internal assessment of the proposed business. By using data from the past and present, descriptive analysis seeks out patterns and connections. Because it describes trends and associations but must look deeper, it is frequently referred to as the most basic form of data analysis. Source of Data The respondent of the study was the owner of Nasnip's Pastry. The researchers chose the respondent because she has more experience operating and managing a pastry shop. In addition, they were the ones who could give accurate data that were needed for the study. Data Gathering Procedure
The data-gathering instrument used in the study was an interview guide. The interview guide was used to gather information and collect accurate data from Nasnip's
Application of the Graphical Method
- Used for solving linear programming models.
- Primarily used for cases with two variables. 3. Although it is not very practical for many variables, it is beneficial for interpreting and analyzing the result and sensitivity of the problem. The algorithm in Solving Graphical Method Step 1: Formulate the Linear programming problem. Step 2: Construct a graph and plot the constraint lines. Step 3: Determine the right side of each constraint line. Step 4: Identify the feasible solution region. Step 5: Plot the objective function on the graph. Step 6: Find the optimum point. Step 7: Calculate the coordinates of the optimum point. Step 8: Determine the value of the objective function using the optimal solution. Application of the Simplex Method
- Used for solving linear programming models.
- Hand using slack variables, tableaus, and pivot variables to find the optimal solution.
- Used to perform row operation on the linear programming model and checking optimal.
The algorithm in Solving Simplex Method Step 1. Convert each inequality in the set of constraints to an equation by adding slack variables. Step 2. Create the initial simplex tableau. Step 3. Locate the most negative entry in the bottom row. The column for this entry is called the entering column. (If ties occur, any tied entries can be used to determine the entering column). Steps 4. Form the ratios of the entries in the “b-column” with their corresponding positive entries in the entering column. The departing row corresponds to the smallest nonnegative ratio (If all entries in the entering column are 0 or negative, then there is no maximum solution. For ties, choose either entry.) The entry in the departing row and the entering column is called the pivot. Steps 5. Use elementary row operations so that the pivot is one and all other entries in the entering column are 0. This process is called pivoting. Steps 6. This is the final tableau if all entries in the bottom row are zero or positive. If not, go back to Step 3. Step 7. If you obtain a final tableau, the linear programming problem has a leading solution, given by the entry in the lower-right corner of the tableau. Standard form of Simplex Method Z= c_1 x_1+c_2 x_2+⋯+c_n x_n Subject to:
RESULTS AND DISCUSSION
This chapter presents the data gathered. The weekly cost of the products, weekly demand per product, and the time required to produce each product were presented in the table and discussed respectively. This demonstrates how the linear programming model was created using the data from Nasnip’s Pastry to maximize its weekly profit. 4.1 DATA FROM NASNIP’S PASTRY Weekly Cost Table 1. Weekly Cost of Nasnip’s Pastry.
Ingredients Weekly Cost
All Purpose Flour 683.
Baking Powder 126.
Baking Soda 70.
Iodize Salt 116.
Cinnamon 91.
Medium Size Ripe Banana 438.
White Sugar 334.
Brown Sugar 225.
Large Egg 560.
Vanilla Extract 35.
Oil 505.
Chocolate Chips 490.
Foil Container (220x115x60MM) 945.
Jersy Evap Milk 343.
Jersy Ube Condensed Creamer 910.
Small Egg 490.
Electricity 1,225.
Transportation 60.
Total 7,646.
Based on Table 1, the total weekly cost of all ingredients of Nasnip’s Pastry is 7646.00. Weekly Average Demand Table 2. Number of Banana and Ube Cake sold weekly from December 12, 2022 to January 8, 2023.
Product 1st 2nd 3rd 4th Total
Banana Cake 39 46 49 43 177
Ube Cake 32 24 19 15 90
The table shows the weekly sales from December 12, 2022, to January 8, 2023. Based on Table 2, Banana Cake is more in demand than Ube Cake. Weekly Time Required Nasnip’s Pastry is open from 6:00 am to 7:00 pm weekly. It offers 13 hours open every day. It uses an electric oven that can fit two cakes in every bake. Every serving of each product, the Banana cake, and Ube cake, requires five minutes for preparation and sixty minutes for baking. All in all, the total weekly time requirement is 5460.00 minutes. 13 ours x 60 minutes x 7 days = 5,460 minutes MODEL FORMULATION Presentation of the Variables Let: z = Weekly profit of Nasnip’s Pastry from Ube Cake and Banana Cake production
