Sample Thesis Paper

SAMPLE THESIS PAPER APPLICATION OF LINEAR PROGRAMMING ON PROMOTION PROBLEMS This thesis deals with the application of linear programming to solving problems in the area of publicity. The emergence of linear programming is a consequence of the development of the so-called operational research. Linear programming is intended for the allocation of scarce resources in order to achieve optimal results. The aim of this thesis is to show that many problems in the field of advertising can be solved by using mathematical programs on the computer. The thesis consists of four chapters. The second chapter will briefly describe the linear programming history, and will list the people who contributed most to linear programming. It will be defined and explained the concept of linear programming. The basic settings of mathematical modeling of problems will be explained, and the areas of possible application of linear programming will be specified. The standard formulation of the linear programming problem will also be displayed. The third chapter will explain the WinQSB computer program and the module for linear and integral linear programming. The most important procedures for resolving mathematical models of economic problems through this program will be described in detail. The input of the input data, the possibility of the subsequent modification of the set conditions, the storage of the model and the results for later use, and the like will be displayed. The fourth chapter will show the resolution of selected examples of advertising problems using the WinQSB computer program, more precisely its module for linear and integral linear programming. The eight selected examples will be elaborated and detailed description of the process of obtaining a mathematical model, as well as the process of  1

SAMPLE THESIS PAPER solving the model using the WinQSB program. Whenever possible, an analysis of the sensitivity of the solution will be carried out by adding new initial conditions. Some useful additional data obtained with the WinQSB computer program will also be interpreted. GENERAL ABOUT LINEAR PROGRAMMING In history, linear programming occurs during the Second World War in planning the cost of equipping military power. Russian mathematician Kantorovich1 introduced the concept of linear programming in 1939 for solving the problem of optimal resource consumption. After the war, research with orientation to organizational issues within the economy continued. In recent times, a special area called operational research has been developed that includes a number of different optimization problems, such as: optimal production programs, optimal inventories, optimum replacements of production assets, optimization of transport, organization, queues, etc. Mathematicians played an important role in the development of linear programming: Dantizig, Foarier, Geuss, Gordon, Minkovski, Farkas and others. Von Neumann had the most important role in relation to many mathematicians regarding the definition and construction of the notion of duality. Linear programming is a mathematical process developed primarily for the needs of analytical support in decision-making processes. In this form it has become one of the most widely used and best known tools in the field of management (Barnett, Ziegler & Byleen, 2005). Linear programming is a mathematical analysis of the problem in which the maximum (minimum) value of the linear function is sought under the given limiting conditions. When observing economic problems, linear systems describe the conditions in which economic processes take place, while the linear function describes a particular demand (goal) that is desired under these conditions. In the process of linear programming, the corresponding systems of equations and inequalities are formed. By solving them with different methods, optimal solutions are obtained. Linear programming addresses a large number of economic problems, and can relate to production, raw materials, labor, market, supply, demand, imports, exports. The primary task of linear programming is to determine the maximum (minimum) linear function depending on a set of set conditions. Some conditions may require, for example, negativity of the variables in the mathematical model and satisfying the constraints written in the form of linear inequalities. In order to deal with the problem of mathematical programming, one must: 2

SAMPLE THESIS PAPER 1. Define the function of goal 2. Form a set of constraints. 3. Choose one or more optimal solutions (Hwang et al., 1980) An economic problem that has these three components can be defined as a problem of mathematical programming. The criterion of the problem of linear programming can be expressed in natural, monetary or other indicators, depending on the nature of the observed problem. In various financial analyzes, a common case is that it is necessary to minimize or maximize some linear function. Most often, this means reducing costs or increasing profits. Of course, desires in this case are directed towards their extremes, i.e. minimum costs and maximum profits. In order to be able to access the search for extremes, one should first define a linear function (objective function) that is generally of the type: min or max z = c1x1 + c2x2 + c3x3 + c4x4 + ….+ cnxn where xj = 1, 2, 3…., n variables called decision-making variables; subject to the following limitations: a11x1 + a12x2 + a13x3 +... + a1nxn ≤ b1 a21x1 + a22x2 + a23x3 +... + a2nxn ≤ b2 a31x1 + a32x2 + a33x3 +...+ a3nxn ≤ b3 Some restrictions can be quite simple (e.g. some decision variables cannot be strictly negative). Other limitation cases may be different and may not be reduced to only the \"be less or equal\" (≤) relationship, but may contain strict equality (=) or the relation \"be greater or equal\" (≥). Each constraint is a linear combination of decision variables. The most common case is that all decision-making choices take non-negative values, so other constraints are added to the so-called natural condition: x1, x2, x3,…, xn ≥ 0. This is the so-called standard formulation of linear programming. It is defined with the goal function, n decision variables and m constraints. The suggestion of specific values of the decision variables is called the solution of the problem of linear programming. If this solution () * * * 1 2,, ..., n x x x satisfies all constraints, it is a permissible solution. The solution is called optimal if it is admissible and minimizes / maximizes the target function. In the case of two variables (n = 2), the graphical method of solving can relatively easily lead to an optimal solution regardless of the number of set limits in the linear programming model. Additional constraints can only increase the number of candidates for the optimal solution, but not the dimension of the space in which the problem is observed. However, in the case of n = 3, the method becomes clumsy as a three-dimensional graphical representation will be required. In the case of n ≥ 4 methods, it is totally unsuitable for use (Chiang, 1996). 3

SAMPLE THESIS PAPER COMPUTER PROGRAM WINQSB WinQSB is a software solution for solving problems in the field of operational research, business decision making and the like. It consists of a total of nineteen application modules. One of the modules is Linear and Integral Linear Programming LP-ILP. This model will be described below, and will be used to address selected examples from the advertising area. In order for the mathematical model of the linear programming problem to be solved using the LP-ILP module, a WinQSB computer program must be started, then click linear and full-line linear programming. This will open a dialog box for entering the initial data on the problem that we want to solve. The corresponding data is entered in the resulting table. In the rectangle next to the Title Problem Title, the name of the problem is typed (for example, Example 1, Problem 1 and the like). In the rectangle next to the Number of Variables, the number of independent variables is entered. In the rectangle next to the Number of constraints, the number of conditions, that is, the constraints that the problem requires is entered. In the Objective Criterion menu, one of the two options offered should be selected depending on the request of the target function: - Maximization if the problem requires maximizing the target function, - Minimization if the problem requires minimization of the target function. In the Data Entry Format menu, one of the two options should also be selected: Spreadsheet Matrix Form if you want to enter the starting data as expanded matrices (as MS Excel spreadsheets), - Normal Model Form if you want to enter the starting data in \"ordinary\" plates, that is, exactly in the form they are placed in a mathematical model. In the Default Variable Type menu one of the four possible types of decision variables should be selected: - Nonnegative continuous if the values of the variables in the mathematical model can be any (positive or negative) decimal numbers, irrational numbers, etc., or if in the model there is at least one variable whose values can be strictly negative, - Nonnegative integer if the values of all variables must be non-negative integers, - Binary if the values of all variables can only be elements of the set {0, 1}. Predefined settings are Maximization, Spreadsheet Matrix Form, and Nonnegative continuous. In practice, the problems of maximizing the target function, whose variables are non-negative real numbers, are most common. Writing the target function and conditions in the matrix form is easier and more transparent than the record in a \"plain\" table form. fter entering the data by clicking \"OK\", a table for entering the coefficients is obtained. In the variable Variable, the names of the independent variables are entered. WinQSB assumes that the variables are labeled with X1, X2, etc. In the Maximize / Minimize line, the target functions must be entered so that the first coefficient is the one with the variable X1, the other one with the variable X2, etc. 4

SAMPLE THESIS PAPER Clicking on the \"File\" option in the main menu opens a drop-down menu that offers options to save the table, search for saved tables, etc. By clicking on the \"Edit\" option in the main menu, a drop-down menu opens, offering options for renaming variables, adding new conditions, deleting existing conditions, adding new variables and deleting existing variables. The troubleshooting option can be obtained by clicking on the \"man\" icon or by clicking on the \"Solve and Problem\" option in the \"Solve and Analyze\" main menu. If an optimal solution is found, a notification appears. Clicking on \"OK\" gives you an optimal solution table. The first column of the output table lists the names of the independent variables. The second column lists the optimal values of independent variables. In the Objective Function line, the optimum value of the target function is printed. . The Constraints column prints the condition mark in the order in which they are entered in the input table. In the Left Hand Side column, the values that are obtained by admitting the optimal values of independent variables are written in each particular condition. In the Right Hand Side column, a value is displayed on the right side of each particular condition. The Slack or Surplus column prints the value differences from the Left Hand Side column and values from the Right Hand Side column. If this difference is equal to 0, the condition is satisfied so that the equality sign is valid. A value other than 0 means that there is a \"surplus\" (if the condition contains a character> =) or \"shortage\" (if the condition contains the character <=). CONCLUSION Linear programming is a relatively young branch of mathematics. It arose during the Second World War. Development has been established to help in many economic problems. This method proved to be excellent for companies that can use this for many savings, more efficient raw materials, more efficient organization of work, and maximization of profit. By achieving these goals, the company is making progress on the market and has an advantage over competition. It is important to know how to use programs and methods that make management easier to make decisions. The thesis presents examples of advertising problems that have been solved using the WinQSB computer program. This program is selected for use because of its simplicity. A user who is not familiar with the work in this program can relatively quickly and easily master the basics of work. Also, the program can be categorized as user-friendly programs (programs tailored to end-users) because in a relatively simple way it enables users to enter data as well as interpret the end results. The selected examples tried at the same time to point out the importance of forming the correct mathematical model, the correct input of data from the model into the computer program, and the interpretation of the results obtained. Particular attention was paid to the analysis of the sensitivity of the results obtained. 5

SAMPLE THESIS PAPER Analytical (\"classical\") analysis is very difficult to carry out, but the computer program enables its quick and easy implementation. The sensitivity analysis of the results obtained is an important factor when making the final decision, as it indicates the \"strength\" of the change in the optimal solution with relatively \"poor\" changes in the initial conditions. Finally, the thesis tried to endorse the thesis that, despite the development of better and faster computer programs, the person as the decision maker can not be replaced. In other words, a person (experts, management, etc.) can not leave a decision on the machine (computer), but can improve existing tools and develop new tools for the purpose of simpler and faster quality business decisions. 6

SAMPLE THESIS PAPER References Hwang, C. L., Paidy, S. R., Yoon, K., Masud, A. S. M. (1980). Mathematical programming with multiple objectives: A tutorial. Computers & Operations Research, Volume 7, Issues 1–2, 1980, Pages 5 - 31. Barnett, R. A., Ziegler, M. R., Byleen, K. E. (2005), Calculus  for Business, Economics, Life Sciences, and Social Sciences, 10th Edition, Pearson Prentice Hall: Upper Saddle River, New Jersey, United States. Chiang, C. A. (1984). Fundamental Methods of Mathematical Economics. McGraw-Hill/ Irwin: New York, United States. 7