PROFIT MAXIMIZATION USING SIMPLEX METHODS: A CASE OF NWENDUS PAINTS COMPANY PORT HARCOURT

PROFIT MAXIMIZATION USING SIMPLEX METHODS: A CASE OF NWENDUS PAINTS COMPANY PORT HARCOURT BY NZEUKWU SALEM CHINECHEREM MAT. NO.: U2018/5575007 A PROJECT SUBMITTED TO THE DEPARTMENT OF MATHEMATICS AND STATISTICS, FACULTY OF SCIENCE UNIVERSITY OF PORT HARCOURT IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF BACHELOR OF SCIENCE (B.Sc) HONS DEGREE IN MATHEMATICS AND STATISTICS JANUARY, 2024 CERTIFICATION I Nzeukwu Salem Chinecherem (U2018/5575007), thereby certify that the work “Profit Maximization Using Simplex Methods: A Case Of Nwendus Paints Company Port Harcourt” was carried out by me under the supervision of Dr. U. C. Orumie, in partial fulfillment of the requirements for the award of bachelor of science (B.Sc) Hons degree in mathematics and statistics, University of Port Harcourt, Rivers State, Nigeria. Dr. U. C. Orumie ___________________ ___________________ (Project Supervisor) Signature Date Dr. J. N. Ezeora ___________________ ___________________ (Head of Department) Signature Date Prof. E. Ehirim ___________________ ___________________ (Dean, Faculty of Science) Signature Date ___________________ ___________________ (External Supervisor) Signature Date DEDICATION This project is dedicated to my lovely parents Rev. and Mrs. Nzeukwu, my Family, and the Lord Almighty. ACKNOWLEDGEMENT I would like to register my profound gratitude to God Almighty for his guidance and grace throughout my life. I also wish to express my deepest gratitude to my supervisor Dr. U. C. Orumie for her gracious efforts and guidance. I would also love to thank my parents, Rev. and Mrs. Nzeukwu, who financially supported my educational pursuit, and my siblings, family, and friends, I love you all. My sincerest gratitude goes to my HOD Dr. J. N. Ezeora and all the staff of the Mathematics and Statistics department for their numerous efforts, I say be blessed, Amen. Abstract The purpose of the study is to highlight the effectiveness of linear programming in the determination of the optimal combination of various products that an organization produces to maximize contribution. For this purpose, data for the month of July 2023 of a paint company in Port Harcourt has been obtained for five product types in 20-liter buckets. The simplex method is used to determine the optimal mix of these paint types to be produced to maximize profit. This was achieved with the help of LINDO 6.1. The results show that the company can earn a maximum profit of #1,240,000.00 weekly by only investing its resources (raw material) in the production of 300 buckets of oil paints and 160 buckets of TEXTCOAT. 1.0 Introduction There is a strict shortage of production inputs that industries are facing nowadays ultimately results in low output. A firm can contribute to the increase in production in the real sector and to economic growth through its managerial decisions about maximum output. Thus managerial decisions are always directed towards finding the right way to maximize profit. Production-oriented industries require optimal management decisions about production levels due to the present competitive pressure. Due to this pressure, several management theories are promulgated to resolve practical problems e.g. industry-specific problems and environmental problems faced by the industries that necessitated the development of several mathematical techniques. Linear programming is one of the most popular and is based on a mathematical approach to reach the optimal solution with constraint resources. There subsist several other mathematical approaches like the OLS model, Markov analysis, time series analysis, and inventory control system. Linear Programming may be termed as a mathematical approach for identifying the optimum combination of products to be produced to maximize profit or minimize cost as per objective within the given resources. Linear programming is also used in operations management to find optimum solutions to several types of problems like allocation of resources, transportation problems, and responsibility problems where linear programming works to choose the best course of action among several alternatives (Yahya, 2004). Linear programming is a term that encompasses mathematical techniques targeted at optimizing outcomes by amalgamating resources (Lucey, 1996). The term “linear” here implies one of the assumptions of linear programming i.e. “linearity of variables”. For being linear by the variables, we mean that they have a straight line or proportional relation. However, the term “programming” refers to an iterative process of moving towards the best solution from the state of “Do nothing”. Linear programming model consists of a linear objective function and several constraint functions. The model is formulated to find the optimum product mix with provided resources (constraints). The objective function in LP model is usually finding the product mix for maximizing contribution or profit or minimizing cost or variable cost whereas, constraint functions are set of restrictions that limit the production at specific level. For example, limited quantity of material availability, limited number of labour hours and limited numbers of machines hours are some of the commonly faced constraints. Moreover, optimal product mix is that point that is achieved with the help of linear programming justifying the constraints and achieving the objective function. Linear programming is extensively used in many areas of study. Most pervasively, it is used to find solution to engineering, economics and business problems. Some organizations where linear programming is used are transportation companies and manufacturing companies etc. LP modelling solves particular mathematical problems by forming specific rules that deals with the allocation of limited resources under strict technological or practical restrictions when certain course of action has to be chosen (Andrade, 1990). LP can represent the model in mathematical form either for minimization (e.g. cost or labour hours) or maximization (e.g. profit or contribution) (Corrar and Teophilo, 2003), (Salau, 1998). Therefore, the researcher wants to apply linear programming technique in Nwendus paint production company to determine the raw material mix that will maximize the company's profit. 1.1 Objective The objectives of this study are: 1. To suggest an optimal product mix that would maximize the contribution of “The company” 2. To highlight the distinctiveness of linear programming modeling at firm level as an optimization technique 3. To contribute in literature about linear programming 1.2 Significance of study This study is significant for showcasing the practical impact of linear programming in optimizing product combinations for profit maximization in a real paint company. Using the simplex method with LINDO 6.1, it highlights the versatility of linear programming in addressing resource allocation challenges and enhancing decision-making in production-oriented industries. The research provides actionable insights, demonstrating the potential for substantial profit gains through optimal product mix decisions. Furthermore, the study contributes to the broader discourse on operations management, showcasing how linear programming can effectively address issues such as resource allocation and transportation logistics. The case study on Nwendus paint production company not only provides practical insights but also serves as a guide for decision-makers, highlighting the potential for significant profit gains through informed product mix decisions facilitated by linear programming techniques. In essence, this research bridges the gap between theoretical understanding and real-world application, reinforcing the strategic significance of linear programming in optimizing industrial processes and driving financial success. 1.3 BRIEF HISTORY OF THE COMPANY Nwendus Paint Company is a paint production company owned by a family and has been in operation since 2008 in ENUGU state. It offers durability and beauty through its color and distributes its product to retailers. It began operating in Port Harcourt in 2015. The company runs in fine segments: Decorative and Architectural furnishes. Apart from paint production, they specialize in color development products advice and paint application. They offer both interior and exterior painting services for commercial and residential projects. the company produces good quality building paints, and automobile paints. CHAPTER TWO 2.0 LITERATURE REVIEW 2.1 Theoretical Background There are contrastive views of researchers about relevance of LP modelling to various managerial decisions. These views are indulged over a long time that improved the LP technique applicability in solving business decision making problems. The economic literature recognize the importance of linear programming particularly for the better planning of developing countries that usually have scarce resources where linear programming assist in apportioning these truncated resources in obtaining optimal solutions. According to Dowing (1992), linear programming is far better than alternative optimization approaches like the Lagrangian method and the Graphic method. Lagrangian approach. In agreement with Turban and Meredith (1991), Dwivedi (2008) claimed that one of the most well-known management science tools is linear programming. The majority of scholars Wagner (2007) and Lucey (2002) studied the idea that linear programming is a method used in operations research. They believe that it is one of the most commercially successful operations research applications, with Wagner (2007) stating that there is strong evidence to suggest that it has the greatest economic impact. Kareem and Aderoba (2008) used data from a cocoa processing plant in Akure, Ondo State, Nigeria, to demonstrate the efficiency of the linear programming model in maintenance and personnel planning. In order to determine the highest profit from the manufacture of soft drinks at the Nigeria Bottling Company, Ilorin facility, Balogun et al. (2012) applied the linear programming method. By using the software linear Interactive Optimizer (LINDO) to maximize the profit of the Khadi and Village Industries Commission (KVIC) connected to Servodaya Sangham, Murugan and Manivel (2009) attempted to illustrate the usage of linear programming technique. The concept of the revised simplex technique with sensitivity analysis was used by Kingmos Paints Nig. Ltd. by Ikpang et al. in 2021. In order to calculate the best profit for the well-known neighborhood bakery Shukura Bakery in Zaria, Kaduna State, Nigeria, Zakariyya et al. (2022) used the linear programming technique evidenced from the history, LP technique is a mathematical tool that was enrooted by George Dantzig, a mathematician, in 1947 for rationalizing logistics for U.S. Air Force. Later, he argued to use of this technique to solve business problems. He also developed the simplex method which was an improved version of linear programming (Dantzig, 1993). In case of allocation problems, there are plenty of activities that are required to be done, being various alternative ways of doing them, with limited resources to be allocated among these activities, then management has to confront the dilemma of how effectively integrate these resources and activities to maximize output. This problem is termed as optimization problem and can be construed with mathematical programming. Charles & Cooper (1963) termed LP modeling as a single objective optimization approach as it exerts to bring out a single objective of either maximization (profit or contribution) or minimization (cost or time). Similarly, Gupta and Hira (2009) argue that LP modeling to optimize a linear function termed as objective function contingent on the aggregation of linear equations known as constraint functions. Dowing (1992) recognized the superiority of linear programming over many other optimization techniques like the Lagrangian method and the Graphic method. Lagrangian method can be used with one constraint, Graphical method can be used with two constraints and linear programming can be used with plenty of constraints. Backing this argument, Dwivedi (2008) conceives that as LP modeling assists in obtaining optimum solutions to constraint problems, it is of inordinate importance for business decision-making. He argued that linear programming requires three particularizations including objective particularization, constraints particularizations, and non-negativity condition. Authenticating this view, many other authors gave the general LP model specification for example Dowling (1992), Henderson et al (2003), Dwivedi (2008), and Koutsoyiannis (1979). Turban and Meredith (1991) also recognize the importance of linear programming as a tool of decision-making in management science. Management science techniques are made of three ingredients: (1) decision variables that cannot be controlled, (2) environmental factors that also cannot be controlled, and (3) result variables. The LP model is also made up of these ingredients with different terms. LP modeling is claimed to be an operations research methodology by many researchers for example by Wagner (2007) and Lucey (2002). They posit that LP is one of the most acknowledged techniques of operations research. Other areas where LP can be used include airlines, energy planning, education, portfolio management, transportation problems and scheduling, etc. According to Sarggeaunt (1965), there are three main problems where linear programming is used including blending and mixing, distribution cost, and planning problems. He also stated the other areas of applicability of linear programming for example decision of plant location, and staff management. An identical list of problems has been provided by Emory and Niland (1968) where linear programming can be productively used. These include the concoction of gasoline stocks, the inception of chemical products, the commingling of fertilizers, and the production of cement, etc. Moreover, Hillier and Lieberman (2001) maintain that apart from allocating resources, there are many other appliances as well. Emory and Niland (1968) cataloged difficulties in adopting linear programming e.g. identification of essential constraints (row equations) and possible choices (column vectors) and presentation of these rows and columns in linear equations. The authentic definition of the problem requires a deep understanding of the LP technique of the company’s operations and the problem to be solved. Another problem may arise in necessary data collection as experiences show that it is always time-consuming and costly to collect data for linear programming. Another problem noted by Turban (1993) is that uncertainties about the behavior of variables like commodity prices make it difficult for managers to choose the best from several alternatives According to Kurtz (1992), linear programming cannot be used in several organizational problems such as employee attitudes and ramifications of strikes. Furthermore, the managers who are proficient in its accurate models also refrain from its use a they are unaware of its applicability. Besides these problems, linear programming has been extensively used in many areas of study. Taha (1992) applied linear programming in poultry farms to determine the accurate proportion of calcium, fiber, and protein in food. Adam et al (1993) implemented linear programming on Multi-Band Enterprises to determine the optimal combination of citizen band radios and portable radios. Murugan and Manivel (2009) applied linear programming on the textile industry with linear interactive optimizer software to determine the optimal numbers of each of three of its products to be produced to maximize the profit of the organization. Adeyemo and Otiero (2009) extended the scope of applicability of linear programming from management sciences to environmental sciences and conducted a study in South African Rand to maximize revenue where sixteen types of crops were implanted in an area of 2500ha. Kareem and Aderoba (2008) attempted to demonstrate the potency of linear programming by applying it in the determination of the optimal crew required for maintenance in Akure (cocoa processing company), Nigeria. The application of linear programming is not restricted to the manufacturing industries, it also works for the service industry e.g. banks. Balbirer (1981) used linear programming to make financial planning of banks (Central Carolina Bank-CCB) to maximize shareholders’ returns. Cohen and Hammer (2013) used linear programming for asset management of banks. Fielitz and Loeffler (2013) implemented linear programming for liquidity management of banks. Guven and Persentili (1997) used linear programming for the balance sheet management of a bank i.e. determining the optimal proportion and combination of assets and liabilities. Majeke (2013) worked on incorporating crop rotational requirements in a linear programming model: A case study of rural farmers in Bindura, Zimbabwe. He used computer software (MS Excel) to solve the models and the result obtained was that the linear programming model maximizes the income of farmers in rural areas. Balogun et al (2012) considered the use of linear programming for optimal production in Coca-Cola Company. They formulated a linear programming model for the production process and solved the model using the simplex algorithm. The result showed that the company should concentrate on the production of the two products in order not to run into high costs. Joly (2012) investigated on the refinery production planning and scheduling. He used the OR technology, and major past and present strategies to deal with present and future challenges. The result obtained was that profitability increase depends on changes in the existing system. CHAPTER THREE 3.0 Methodology To apprehend optimal combinations from various categories of Plastic Emulsion, a Simplex method of solving Linear programming is used. 3.1 LINEAR PROGRAMMING Linear programming is a mathematical modeling technique in which a linear function is maximized or minimized when subjected to various constraints. This technique has been useful for guiding quantitative decisions in business planning, in industrial engineering, and—to a lesser extent—in the social and physical sciences. The solution of a linear programming problem reduces to finding the optimum value (largest or smallest, depending on the problem) of the linear expression (called the objective function) Given these definitions in section 2.0 above, the LP problem is to choose X1 , X2 ,…, Xn so as to Max Z= c1X1 + c2X2 + c3X3 …+cnXn s.t. a11X1 + a12X2 + a13X3 … +a1nXn ≤ b1 a21X1 +a22X2 +a23X2 … +a2Xn ≥ b2 a31X1 +a32X2 + a33X3 … +a3nXn = b3 : : : … : : am1X1 + am2X2+ am3X3 + amnXn≤ bm X1 , X2 , X3,… Xn ≥ 0 Mathematically, an LP model can generally be represented thus; (2.1) such that (2.2) (2.3) This formulation may also be expressed in matrix notation. Max Z = C TX S.t AX ≤ B X ≥0 Equation (2.1) is the linear objective function to be optimized (i.e. to maximize or minimize). It is expressed in terms of decision variables to optimize the criterion of optimality also called (measure-of-performance). Where Z is a measure-of-performance variable which is a function of . The quantities are parameters that represent the contribution of a unit of the respective variable to the measure-of-performance i.e. (cost coefficient). Equation (2.2) consists of n linear constraints in which each constraint specifies the exact (for =) or lower (for ) or upper (for ) boundary limit of each available resource. There are always certain limitations on the use of resources that limit the degree to which an objective can be achieved. Such constraints must be expressed as linear equalities or inequalities in terms of decision variables. The solution of an LP model must satisfy these constraint equations. Equation (2.3) represents non-negative constraints which imply that negative values of the decision variable have no practical meaning. The coefficients aij for i= 1, 2, ..., m, j =1, 2, . . . ,n are called the technological coefficient and represents the coefficient of the j th decision variable in i th constraints. is referred to as the right-hand side value of the i th constraints. xj ,j=1,2….n is the j th decision variables which are variables whose values are to be determined in some optimal fashion. is the number of linear constraints while represents the number of decision variables. The basic assumption in the application of this method is that the various relationships between demand and availability are linear; that is, none of the xi is raised to a power other than 1. To obtain the solution to this problem, it is necessary to find the solution of the system of linear inequalities (that is, the set of n values of the variables xi that simultaneously satisfies all the inequalities). The objective function is then evaluated by substituting the values of the xi in the equation that defines z. 3.1.1 ASSUMPTIONS OF LINEAR PROGRAMMING To represent an optimization problem as a linear program, several assumptions that are implicit in the linear programming formulation are needed. A brief discussion of these assumptions is given below. The first three involve the appropriateness of the formulation, whereas the last four assert the mathematical relationships within the model. Objective Function Appropriateness This assumption means that within the formulation the objective function is the sole criterion for choosing among the feasible values of the decision variables . The objective to be achieved must be simple. i.e. Decision Variable Appropriateness A key assumption is that the specification of the decision variables is appropriate. This assumption requires that; a) The decision variables are all fully manipulatable within the feasible region and are under the control of the decision maker. b) All appropriate decision variables have been included in the model. Constraint Appropriateness The third appropriateness assumption involves the constraints. Again, this is best expressed by identifying sub-assumptions: a) The constraints fully identify the bounds placed on the decision variables by resource availability, technology, the external environment, etc. Thus, any choice of the decision variables, which simultaneously satisfies all the constraints, is admissible. . b) The resources used and/or supplied within any single constraint are homogeneous items that can be used or supplied by any decision variable appearing in that constraint. c) Constraints have not been imposed which improperly eliminate admissible values of the decision variables. d) The constraints are inviolate. No considerations involving model variables other than those included in the model can lead to the relaxation of the constraints. Proportionality/linearity Proportionality requires the contribution per unit of each decision variable to the objective function and its requirements in the constraints to be directly proportional to the value of the variable. This contribution is assumed constant and independent of the variable level. Similarly, the use of each resource per unit of each decision variable is assumed constant and independent of variable level. Linearity on the other hand states that a linear relationship must exist between the resources consumed and the contribution margin. i.e. the amount of each resources used (or supplied) and its contribution to the objective function must be proportional to the value of each decision variable. Additivity Additivity deals with the relationships among the decision variables. Simply put, their contributions to an equation must be additive. The total value of the objective function equals the sum of the contributions of each variable to the objective function. Similarly, total resource use is the sum of the resource use of each variable. This requirement rules out the possibility that interaction or multiplicative terms appear in the objective function or the constraints. Divisibility The problem formulation assumes that all decision variables can take on any non-negative value including fractional ones. This assumption is violated when non-integer values of certain decision variables make little sense. For instance, a decision variable may correspond to the purchase of a tractor or the construction of a building. Such therefore must be assigned an integer value. No negativity of the variables For most practical problems, the variables represent physical quantities and hence must be non-negative. This assumption states that a negative unit of production cannot be achieved, but it is possible not to produce a particular product line at the expense of others, if that decision will optimize the total contribution of the organization. i.e . An optimization problem can be cast as a linear program only if the aforementioned assumptions hold. Applications of the method of linear programming were first seriously attempted in the late 1930s by the Soviet mathematician Leonid Kantorovich and by the American economist Wassily Leontief in the areas of manufacturing schedules and of economics, respectively, but their work was ignored for decades. During World War II, linear programming was used extensively to deal with transportation, scheduling, and allocation of resources subject to certain restrictions such as costs and availability. These applications did much to establish the acceptability of this method, which gained further impetus in 1947 with the introduction of the American mathematician George Dantzig’s simplex method, which greatly simplified the solution of linear programming problems. However, as increasingly more complex problems involving more variables were attempted, the number of necessary operations expanded exponentially and exceeded the computational capacity of even the most powerful computers. Then, in 1979, the Russian mathematician Leonid Khachiyan discovered a polynomial-time algorithm—in which the number of computational steps grows as a power of the number of variables rather than exponentially—thereby allowing the solution of hitherto inaccessible problems. However, Khachiyan’s algorithm (called the ellipsoid method) was slower than the simplex method when practically applied. In 1984 Indian mathematician Narendra Karmarkar discovered another polynomial-time algorithm, the interior point method, that proved competitive with the simplex method. 3.1.2 Simplex method The simplex method is, a standard technique in linear programming for solving an optimization problem, typically one involving a function and several constraints expressed as inequalities. The inequalities define a polygonal region, and the solution is typically at one of the vertices. The simplex method is a systematic procedure for testing the vertices as possible solutions. Some simple optimization problems can be solved by drawing the constraints on a graph. However, this method is useful only for systems of inequalities involving two variables. In practice, problems often involve hundreds of equations with thousands of variables, which can result in an astronomical number of extreme points. In 1947 George Dantzig, a mathematical adviser for the U.S. Air Force, devised the simplex method to restrict the number of extreme points that have to be examined. The simplex method is one of the most useful and efficient algorithms ever invented, and it is still the standard method employed on computers to solve optimization problems. First, the method assumes that an extreme point is known. (If no extreme point is given, a variant of the simplex method, called Phase I, is used to find one or to determine that there are no feasible solutions.) Next, using an algebraic specification of the problem, a test determines whether that extreme point is optimal. If the test for optimality is not passed, an adjacent extreme point is sought along an edge in the direction for which the value of the objective function increases at the fastest rate. Sometimes one can move along an edge and make the objective function value increase without bound. If this occurs, the procedure terminates with a prescription of the edge along which the objective goes to positive infinity. If not, a new extreme point is reached having at least as high an objective function value as its predecessor. The sequence described is then repeated. Termination occurs when an optimal extreme point is found or the unbounded case occurs. Although in principle the necessary steps may grow exponentially with the number of extreme points, in practice the method typically converges on the optimal solution in a number of steps that is only a small multiple of the number of extreme points. 3.1.2.1 Simplex algorithm The Simplex method is an approach to solving linear programming models by hand using slack variables, tableaus, and pivot variables as a means to finding the optimal solution of an optimization problem. A linear program is a method of achieving the best outcome given a maximum or minimum equation with linear constraints. Most linear programs can be solved using an online solver such as MatLab, but the Simplex method is a technique for solving linear programs by hand. To solve a linear programming model using the Simplex method the following steps are necessary: ● Standard form ● Introducing slack variables ● Creating the tableau ● Pivot variables ● Creating a new tableau ● Checking for optimality ● Identify optimal values Step 1: Standard Form Standard form is the baseline format for all linear programs before solving for the optimal solution and has three requirements: (1) must be a maximization or minimization problem, (2) all linear constraints must be in a less-than-or-equal-to or greater than or equal to inequality, (3) all variables are non-negative. These requirements can always be satisfied by transforming any given linear program using basic algebra and substitution. Standard form is necessary because it creates an ideal starting point for solving the Simplex method as efficiently as possible as well as other methods of solving optimization problems. Step 2: Determine Slack or surplus Variables Slack or surplus variables are additional variables that are introduced into the linear constraints of a linear program to transform them from inequality constraints to equality constraints. If the model is in standard form, the slack or surplus variables will always have a +1 or a -1 coefficient respectively. Slack or surplus variables are needed in the constraints to transform them into solvable equalities with one definite answer. After the slack variables are introduced, the tableau can be set up to check for optimality as described in Step 3. Step 3: Setting up the Tableau A Simplex tableau is used to perform row operations on the linear programming model as well as to check a solution for optimality. The tableau consists of the coefficient corresponding to the linear constraint variables and the coefficients of the objective function. The following two rows represent the linear constraint variable coefficients from the linear programming model, and the last row represents the objective function variable coefficients. Once the tableau has been completed, the model can be checked for an optimal solution as shown in Step 4. Step 4: Check Optimality The optimal solution of a maximization linear programming model is the values assigned to the variables in the objective function to give the largest zeta value. The optimal solution would exist on the corner points of the graph of the entire model. To check optimality using the tableau, all values in the last row must contain values greater than or equal to zero. If a value is less than zero, it means that the variable has not reached its optimal value. As seen in the previous tableau, three negative values exist in the bottom row indicating that this solution is not optimal. If a tableau is not optimal, the next step is to identify the pivot variable to base a new tableau on, as described in Step 5. Step 5: Identify Pivot Variable The pivot variable is used in row operations to identify which variable will become the unit value and is a key factor in the conversion of the unit value. The pivot variable can be identified by looking at the bottom row of the tableau and the indicator. Assuming that the solution is not optimal, pick the smallest negative value in the bottom row. One of the values lying in the column of this value will be the pivot variable. To find the indicator, divide the beta values of the linear constraints by their corresponding values from the column containing the possible pivot variable. The intersection of the row with the smallest non-negative indicator and the smallest negative value in the bottom row will become the pivot variable. Now that the new pivot variable has been identified, the new tableau can be created in Step 6 to optimize the variable and find the new possible optimal solution. Step 6: Create the New Tableau The new tableau will be used to identify a new possible optimal solution. Now that the pivot variable has been identified in Step 5, row operations can be performed to optimize the pivot variable while keeping the rest of the tableau equivalent. I. To optimize the pivot variable, it will need to be transformed into a unit value (value of 1). To transform the value, multiply the row containing the pivot variable by the reciprocal of the pivot value. II. After the unit value has been determined, the other values in the column containing the unit value will become zero. III. In order to keep the tableau equivalent, the other variables not contained in the pivot column or pivot row must be calculated by using the new pivot values. For each new value, multiply the negative of the value in the old pivot column by the value in the new pivot row that corresponds to the value being calculated. Then add this to the old value from the old tableau to produce the new value for the new tableau. This step can be condensed into the equation on the next page: New tableau value = (Negative value in old tableau pivot column) x (value in new tableau pivot row) + (Old tableau value) Step 7: Check Optimality As explained in Step 4, the optimal solution of a maximization linear programming model are the values assigned to the variables in the objective function to give the largest zeta value. Optimality will need to be checked after each new tableau to see if a new pivot variable needs to be identified. A solution is considered optimal if all values in the bottom row are greater than or equal to zero. If all values are greater than or equal to zero, the solution is considered optimal and Steps 8 through 11 can be ignored. If negative values exist, the solution is still not optimal and a new pivot point will need to be determined which is demonstrated in Step 8. Conclusion The Simplex method is an approach for determining the optimal value of a linear program by hand. The method produces an optimal solution to satisfy the given constraints and produce a maximum zeta value. To use the Simplex method, a given linear programming model needs to be in standard form, where slack variables can then be introduced. Using the tableau and pivot variables, an optimal solution can be reached. 2 Standardization and Initialization of simplex method tableau Consider the generalized linear programming problem in (2.1) and (2.2), then to standardize, add slacks or subtract surpluses to obtain; Z - =0 (2.4) s.t (2.5) This can be represented thus; And becomes; Represented in a simplex tableau as; Row z 1 . . . 1 2 . . . BV RHS 0 1 − 1 . . . − 0 0 . . . 0 z = 0 1 0 − 11 . . . − 1 1 0 . . . 0 = 1 1 2 0 − 21 . . . − 2 0 1 . . . 0 2 2 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ m 0 . . . 0 0 . . . 1 1 = Then, algorithm starts with an initial basic feasible solution (bfs) and tests its optimality. If optimality condition is reached, then the algorithm terminates. Otherwise, the algorithm identifies an adjacent bfs, with a better objective value. The optimality of this new objective is tested again, and the entire approach is repeated, until an optimal bfs is found. Since every time a new bfs is identified, the objective value is improved, and the set of bfs is finite, it follows that the algorithm terminates in a finite number of steps (iterations). In order words, the simplex method is defined as an iterative process in which we start with the solution , that satisfies the equation (2.1)-(2.5), then look for a new solution , which is better in the sense that it has a larger (smaller) objective function value in maximization (minimization) case, and the process continues until a solution that cannot be improved further is arrived at. This final solution is then the optimal solution. By introducing the idea of slack variables, we add more variables to the problem. So we need to use the simplex method to solve this more complex problem. Definition of Terms Basic variables are variables that are non-negative in terms of the optimal solution. Constraints are a series of equalities and inequalities that are a set of criteria necessary to satisfy when finding the optimal solution. Inequality is an expression that does not have one definite solution and is distinguishable by its ‘greater than’ or ‘less than’ symbols in the place of a traditional equal sign. Linear program is a model used to achieve the best outcome given a maximum or minimum equation with linear constraints. Non-basic variables are variables that are zero in terms of the optimal solution. Optimal solution of a maximization linear programming model are the values assigned to the variables in the objective function to give the largest zeta value. The optimal solution would exist on the corner points of the graph of the entire model. Pivot variable is used in row operations to identify which variable will become the unit value and is a key factor in the conversion of the unit value. Simplex method is an approach to solving linear programming models by hand using slack variables, tableaus, and pivot variables as a means to finding the optimal solution of an optimization problem. Simplex tableau is used to perform row operations on the linear programming model as well as for checking optimality. Slack variables are additional variables that are introduced into the linear constraints of a linear program to transform them from inequality constraints to equality constraints. Standard form is the baseline format for all linear programs before solving for the optimal solution. 3.2 DATA COLLECTION AND DATA PRESENTATION For the purpose of accomplishing these objectives, data has been collected from a renowned paint manufacturing company. The company produces various types of paints. The records of company show that the product names are SILK, MATT, OIL PAINTS, EMULSION, TEXT COAT are the main product in terms of number of units to be sold and thus generating contribution with six major raw materials available for weekly production and the profit contribution per bucket of paint produced. These products are made in same sizes (20 litres). The data used in the study is a one week production in the month of July 2023 obtained from company records and from interviews from Manufacturing department personal and Accounts department personnel. The individual raw material per each unit product of paint produced is shown as follows. The data collected is tabulated below SILK, MATT OIL PAINTS EMULSION TEXT COAT AVAILABILITY Water in liters 4 7 10 8 2000 calcium 5 15 15 2400 Resine 10 5 2 1.5 2.5 1000 Tickner 0.1 0.1 0.1 0.1 30 Preservative 0.1 0.1 0.1 0.1 29 Profit per unit product in naira 4000 3000 2000 2000 4000 Linear programming converts the data into objective function (in terms of contribution per unit) and relevant constraints functions (in terms of material quantity per unit) which are as follows: Let x1,x2,x3,x4 and x5 represents 20 liters of SILK, MATT, OIL PAINTS, EMULSION, and TEXT COAT respectively. Then the objective function becomes (z) = 4000x1 + 3000x2 + 2000x3+ 2000x4+4000x5 S.T The raw material constraints are formed below: 4x1 + 7x2 + 0x3 + 10x4 + 8x5 ≤ 2000 Water 0x1 + 5x2 + 0x3 + 15x4 + 15x5 ≤ 2400 calcium 10x1 + 5x2 + 2x3 + 1.5x4 + 2.5x5 ≤ 1000 Resine 0.1x1 + 0.1x2 + 0x3 + 0.1x4 + 0.1x5 ≤ 30 Tickner 0.1x1 + 0.1x2 + 0x3 + 0.1x4 + 0.1x5 ≤ 29 Preservative x1 , x2 , x3 , x4, x5>=0 Where coefficients represent the quantity of material to be used to manufacture each product and constant in right-hand side represents maximum quantity available of each type of material. The first step in the simplex method is the introduction of slack variables and converting inequalities into equalities. Converting the above LPP into standard form gives: z -4000x1 - 3000x2 - 2000x3- 2000x4-4000x5=0 S.T 4x1 + 7x2 + 0x3 + 10x4 + 8x5 + 1 = 2000 0x1 + 5x2 + 0x3 + 15x4 + 15x5 + 01 + 2 = 2400 10x1 + 5x2 + 2x3 + 1.5x4 + 2.5x5 + 01 + 02 + 3 = 1000 0.1x1 + 0.1x2 + 0x3 + 0.1x4 + 0.1x5 + 01 + 02 + 03 + 4 = 30 0.1x1 + 0.1x2 + 0x3 + 0.1x4 + 0.1x5 + 01 + 02 + 03 + 04 + 5 = 29 x1 , x2 , x3 , x4, x5, >= 0 The above standardized form will be represented on the initial Tablue. Table 3.1: Initial Tablue Z 1 2 3 4 5 1 2 3 4 RHS Z 1 4000 3000 2000 2000 4000 0 0 0 0 0 s1 0 4 7 0 10 8 1 0 0 0 0 2000 s2 0 0 5 0 15 15 0 1 0 0 0 2400 s3 0 10 5 2 1.5 2.5 0 0 1 0 0 1000 s4 0 0.1 0.1 0 0.1 0.1 0 0 0 1 0 30 s5 0 0.1 0.1 0 0.1 0.1 0 0 0 0 1 29 CHAPTER FOUR DATA ANALYSE, SUMMARY AND CONCLUSIONS 4.1 ANALYSES Solving this problem using LINDO DATA INPUT RESULT OUTPUT ● LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE 1) 1240000. VARIABLE VALUE REDUCED COST X1 0.000000 6000.000000 X2 0.000000 2500.000000 X3 300.000000 0.000000 X4 0.000000 1000.000000 X5 160.000000 0.000000 ROW SLACK OR SURPLUS DUAL PRICES 2) 720.000000 0.000000 3) 0.000000 100.000000 4) 0.000000 1000.000000 5) 14.000000 0.000000 6) 13.000000 0.000000 NO. ITERATIONS = 2 4.3 INTERPRETATION OF THE RESULT/ The output shows that the optimum solution occurred at the 2 nd iteration with the values as shown in the table above. It shows that the company can make up to #1,240,000.00 weekly if they continue to produce 300 buckets of X3(OIL PAINTS) and 160 buckets of X5(TEXT COAT) weekly without going into production of any of x1(SILK), x2(MATT), and x4 (EMULSION). The slack variables in row 2 and row 5 show that there is so much wastage of materials in the production of x1, x2, and x4 (SILK), (MATT), and (EMULSION) respectively. 4.4 Summary and Conclusion The study is conducted on NWENDUS paint company in Port Harcourt. The purpose of the study is to determine the optimal quantity of five different types of paints to be produced namely; SILK, MATT, OIL PAINTS, EMULSION, TEXT COAT (in 20 litres bukets). For this purpose, the Simplex Method is used for this analysis with the help of LINDO 6.1. The results of the simplex method show that the company can earn a maximum profit of #1,240,000.00 weekly by producing only 300 buckets of X3(OIL PAINTS) and 160 buckets of X5(TEXT COAT) all in 20litres. The values of slack are resources left after production. Recommendations The researcher therefore recommends that slack values which are the leftover resources should be put into other projects of the company or put into more production of the two product types to increase their income. These findings would be helpful for the company to maximize its contribution not only for this month but the company can use this technique throughout its existence. 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