Maximizing Profits in Manufacturing: Nasnip's Pastry Case Study with Linear Programming, Thesis of Mathematics

Download Maximizing Profits in Manufacturing: Nasnip's Pastry Case Study with Linear Programming and more Thesis Mathematics in PDF only on Docsity! 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 2 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. 5 This study aims to maximize Banana Cake's and Ube Cake's weekly profit on Nasnip's Pastry. Specifically, it will determine the following.  1. To determine the weekly cost per product, the demand per product, and the time requirement to produce each product. 2. To establish a model based on the data gathered. 3. 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. REVIEW OF LITERATURE This chapter presented the related literature and studies gathered from different thesis, and internet that have bearing with the present study. All these works have contributed to a great extend in the realization of the present study. Review of Related Literature According to The Economic Times (2023), Business refers to an enterprising entity or organization that carries out professional activities. They could be industrial, commercial, or other. For-profit businesses operate to make a profit, whereas nonprofit ones do so to further a charitable mission. Examples of business ownership include partnerships, single proprietorships, corporations, and others. There are both small-scale and large-scale businesses with Amazon and Walmart as two examples of large corporations worldwide. According to McDaniel (2018), the output of goods and services that people can purchase with their available funds indicates a nation's standard of living. The products and services that serve as the basis of our standard of living are thus produced by businesses. One of the highest standards of living can be found in the United States. Although many nations, including Switzerland and Germany, have average incomes more excellent than those in the United States, their living standards are not higher due to the high cost of living. Therefore, the exact quantity of money has a lower value in those nations. Gilman (2018) explained that a business strives for profit by providing goods and services its customers desire. It is the efforts and activities of a person producing goods or 7 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 10 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, . . 11 ., 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: 12 1. Graphical Method: This method is used to optimize the two-variable linear programming. If the problem has two decision variables, the graphical method is the best method to find the optimal solution. 2. Simplex Method: This method is used for solving an optimization problem involving a function and several constraints for problems with more independent variables. Revised Simplex Method: This is technically equivalent to the traditional simplex method but is implemented differently. Graphical Method            In order to identify the best (optimal) solution for LP problems with only two variables, it is possible to graphically display the entire set of feasible solutions by plotting linear constraints on the graph. The term "graphical solution method" means finding the best solution to an LP problem with two variables (Murray, 2006).  The graphical method uses to solve linear optimization problems involving two variables. When two variables are in the problem, it can refer to them as x1 and x2 and do most of the analysis on a two-dimensional graph. (Geektonight, 2022). Graphical methods provide a visual presentation of linear programming algebra, and they can anchor the understanding of basic definitions and possibilities. For these reasons, the graphical approach provides valuable background for working with linear programming concepts. However, solving a linear programming problem by drawing its graph becomes challenging when three or more variables are involved (Geektonight, 2022). 14 Pastry. It only contains a few topics to cover in the interview. It includes the total cost per week, the product material, demand per product, and the production time requirements. The interview guide was used as a suggestive reference or assistant during the interview. The researchers first identified and selected a respondent and made a letter of intent and interview guide which formulated some questions about Nasnip's Pastry. Then, the researchers prepared for preliminary visits to the bakery's location and the selected respondent. Therefore, a letter of permission with an interview guide to conduct an interview was sent to them, setting an appointment to avoid conflict in their work schedules. Upon approval, the interview was conducted at their most convenient time. Chosen respondents were asked questions from the interview guide. After the interview, all the data gathered were used as the guide and the needed information to accomplish and finish the study. Tools for Data Analysis With a requirement for the advanced level being the use of technology. The researchers used web-based software applications to explore data. The Desmos software provides a most valuable addition to the study. It works well with datasets; data can be typed directly into Desmos. The calculator can plot a variety of different equations and from lines to parabolas and more. It can take a set of data and create a graph needed to study to get the feasible region. Furthermore, the researchers used eMH (eMathHelp), a website calculator that solves various calculus problems. In addition, as a guide to getting an accurate solution to the linear programming problem. 15 Application of the Graphical Method 1. Used for solving linear programming models. 2. 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 1. Used for solving linear programming models. 2. Hand using slack variables, tableaus, and pivot variables to find the optimal solution. 3. Used to perform row operation on the linear programming model and checking optimal. 16 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: 19 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 20 Decision Variables Let:            x = number of Banana cakes to be made weekly            y = number of Ube cakes to be made weekly OBJECTIVE FUNCTION Table 3. Nasnip’s Products and Prices. Product Price Banana Cake 130.00 Ube Cake 130.00  Table 3 shows that Nasnip’s Pastry sells two kinds of cakes. The price of the cakes is PHP 130.00. Sales = Price of Banana cake (x) + Price of Ube cake (y) Sales = 130x + 130y Then, the objective function is: Max Profit = Sale - Cost Max z = 130x + 130y – 7646.00 This study aims to maximize the weekly profit on products of Nasnip’s Pastry. This can be achieved by determining the weekly cost per product, the weekly demand per product, and the weekly time required to produce each product. The profit can be found by subtracting the weekly sale from the weekly cost of the product. 7646.00 represents the weekly cost of Nasnip’s Pastry (Table 1). 21 Cost Constraints 104.00x + 111.00y ≤ 9100.00 Table 4. Nasnip’s Pastry cost of Banana Cake per serving. Ingredients Weight (g) Cost All Purpose Flour 95 6.08 Baking Powder 2.5 0.9 Baking Soda 5 1.00 Iodize Salt 5 1.09 Cinnamon 2.5 1.3 Medium Size Ripe Banana 156.25 6.25 White Sugar 47.5 4.75 Brown Sugar 45 3.6 Large Egg 1pc. 8.00 Vanilla Extract 5 0.48 Oil 40 3.75 Chocolate Chips 16 6.8 Jersey Evaporated Milk 50 4.05 Foil Container (220x115x60MM) 5pcs. 9.00 Transportation Back and forth 30.00 Electricity 60 minutes (1.2KWH) 16.99 Total 104.00 Table 5. Nasnip’s Pastry cost of Ube Cake per serving. Ingredients Weight (g) Cost All Purpose flour 112.5 7.2 Baking Powder 5 1.8 Iodize Salt 5 1.09 Oil 40 3.75 Small Egg 2pcs. 14.00 Jersy Ube Condensed Creamer 195 26.00 Jersy Evaporated Milk 15 9.00 Foil Container 1pc. 9.00 Transportation Back and forth 30.00 Baking Time 60 minutes 16.99 Total 111.00 24 Max z = 130x + 130y - 7646 Subject to: 104x + 111y ≤ 9100 (65) (x/2) + (65) (y/2) ≤ 5460 x + y ≤ 168 x ≥ 44 y ≥22 x, y ≥ 0 OPTIMAL SOLUTION (GRAPHICAL METHOD) After planning the activities to obtain an optimal best and expected result among feasible alternatives, the next step involving the problem is to use the graphical method. Figure 1: Feasible Region (desmos) 25 By inserting the constraints into Desmos (graphical calculator), it shows the result of the graph. It determines the feasible region, the small triangle bounded by the intersection points of the boundary lines and the corner points. The corner points are: (0,0) (44, 22) (44, 40.757) (64.019, 22) By evaluating each corner points to the objective function, the result satisfies all the constraints and shows that the maximum profit can be achieved at (64.019, 22) with PHP 3,536.47 profit. OPTIMAL SOLUTION (SIMPLEX METHOD) The researchers use linear programming: Simplex method to compare the result to the Graphical method. The researchers used eMH (eMathHelp), a website calculator that solves various calculus problems. In addition, as a guide to getting an accurate solution to the linear programming problem. Solution: All variables need to make non-negative; therefore, replace all unrestricted variables x i with x i +¿−W ¿, add slack variables to turn all the inequalities into equalities, add artificial variables, and penalize the artificial variables in the objective function: z=130 x+¿+130 y+¿−260W −M Y 1−M Y 2→max ¿¿ Subject to: 104 x+¿+111 y+¿−215W +S1=9100¿¿ 26 65¿ x+¿+ y+¿−2W +S3=168 ¿¿ x+¿−W −S4+Y1=44 ¿ y+¿−W −S5+Y2=22 ¿ x+¿, y+¿,W ,S1,S2 ,S3 ,S4, S5,Y 1,Y 2 ≥ 0¿ ¿ Simplex Tableau: Basic x+¿¿ y+¿¿ W S1 S2 S3 S4 S5 Y 1 Y 2 Solution S4 0 0 0 1 104 0 0 1 111 104 −1 −111 104 1041 52 S2 0 0 0 −5 16 1 0 0 −35 16 0 35 16 21315 8 S3 0 0 0 −1 104 0 1 0 −7 104 0 7 104 4263 52 x+¿¿ 1 0 −1 1 104 0 0 0 111 104 0 −111 104 3329 52 Y 2 0 1 −1 0 0 0 0 −1 0 1 22 z 0 0 0 5 4 0 0 0 35 4 M M−35 4 22365 2 The optimum has been reached. Optimal values; x +¿=3329 52 ¿, y+¿=22 ¿, W =0, S1=0, S2= 21315 8 , S3= 4263 52 , S4= 1041 52 , S5=0, Y 1=0, Y 2=0. z = 3536.5 is achieved at ( 3329 52 , 22) ≈(64.019, 22). Evaluating the result of the constraints and objective function satisfies all the constraints and shows that the maximum profit can be achieved at (64.019, 22) with PHP 3,536.47 profit. After getting the outcomes of the two methods, the researchers compared the Graphical Method and Simplex Method results. It was determined through comparison that the graphical method is accurate because the result corresponds to the simplex 29 to be made weekly. The weekly profit was also represented by the variable “z”. In this study, the model of linear programming has different kinds of constraints, which are the cost, the demand, the time, the oven capacity, and the non-negative constraints. The constraints for the cost were constructed based on the ingredients of Banana cake and Ube cake weekly. The demand constraints were constructed by the sales for December 12, 2022, to January 8, 2023, in 4 weeks. The linear programming was solved using Desmos, a graphical calculator that helps to find accurate corner points and feasible regions, and Emh. This software calculator solves a wide range of calculus problems. Based on the analysis of both linear programming methods, the results show that the value of the two decision variables is 64 for Banana cake and 22 for Ube cake. The maximum value of “z” or weekly profit is PHP 3536.47 weekly. Conclusion Maximization is the process of business firms to ensure the best output and price level are achieved to maximize return. It is the mathematical process of finding the maximum value of a function. The graphical method is the easiest linear programming solution, usually finding a solution quickly. This study demonstrates that this method is suitable for achieving the maximum profit for Nasnip’s Pastry. The following conclusion shows the limitation of profit in Nasnip's Pasty. First, Nasnip's Pastry has a minimal oven capacity. This problem limits Nasnip's Pastry's ability to bake more cakes per hour. Based on that, Nasnip's Pastry could invest in a larger oven to bake more products. It will help them to maximize their time by baking more in the 30 required time, and in that case, it would lower the cost of electricity. They also need to raise their weekly budget to cover the weekly cost and maximize the weekly time required to make their products to gain profit. Nasnip's Pastry may decrease the production of Ube cake; based on the total weekly expenses, Ube cake was much higher than Banana cake; however, Banana cake has fewer expenses and can give more profit than Ube cake. Moreover, this study has shed light on the profitability of using Linear Programming techniques. Graphical Method and Simplex Method satisfy all criteria and provide the highest profit by evaluating each corner point's contribution towards the objective function. Based on the data obtained and the result shown by both methods, it was discovered that Nasnip's Pastry should produce and sell sixty-four pieces of Banana cake and twenty-two pieces of Ube cake weekly, respectively, in order to achieve the maximum weekly profit of PHP 3536.47. Recommendation  The solution satisfies all criteria of linear programming techniques. In order to allocate limited profit toward their full potential, optimization techniques are beneficial. The following recommendations are suggested: The researchers recommend that Nasnip's Pastry raise its weekly budget to maximize the weekly time required to make its product to gain a higher profit. It is crucial to make the business budget work to enter the correct numbers and make the right decisions to use it to track your expenses and where there is much spending, like the product of Ube cake, based on the total expenses. Ube cake was much higher than 31 Banana cake. However, Banana cake has fewer expenses and can give more profit than Ube cake. Proper budgeting can increase profits by cutting unnecessary expenses. Nasnip's Pastry should also consider investing to buy a bigger oven for them to bake more products. It will help them to maximize their time by baking more in the required time. Since we are in the modern technologies the researchers also suggest that Nasnip's Pastry must recognize making other strategies such as having an online engagement like posting on social media, making order online and market strategy. Social media will be the best platform to promote, it plays a critical role in marketing their business if used effectively, and the opportunity to reach and engage with such a huge audience. Social media is a great way to connect with people who already engage with your products and introduce the business to people who are yet to discover Nasnip’s Pastry. Also Invest in photography and video and create a consistent brand identity with your assets. It also worth knowing that the video is one of the most engaging assets, so use it whenever possible. Give people what they want, instead of following trends determine what best serves your buyer. It’s okay to be contrary to what’s currently popular. To offer services your buyers want (and thus making sure they come back to you and tell others about you), then improve the offer so it’s better than the rest. Build a plan that will iterate and evolve. And lastly understand that succession is part of the evolution of any organization and business, change is constant and people don’t deal with change well. Embrace the journey and don’t treat it like it is “negative” or an “event”. It is a fantastic opportunity to evolve the business and to ensure it thrives (not just survives) in the future.