Linear Programming & Optimizing the Resources

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1 Linear Programming & Optimizing the Resources Abstract Maryam Solhi Lord Samira Mohebbi Bazardeh Sharareh Khoshnood Nastaran Mahmoodi Fatemeh Qowsi Rasht-Abadi Marjan-ol-Sadat Ojaghzadeh Mohammadi MA Students in Commercial Management (International Marketing), Islamic Azad University, Rasht, Iran Nowadays, managers are evaluated by their decision-making. Linear programming is one of the strongest techniques which can be used by managers to solve problems considering/subject to the settings of the problem. By applying the linear programming, the managers are trying to maximize their profit on one hand, and minimize their costs on the other. In this paper, the previous research background has been explained, followed by a discussion on the linear programming and its applications. Keywords: MAXIMIZING, SIMPLEX, OBJECTIVE FUNCTION, LINEAR CONSTRAINT. Introduction Linear programming, or linear optimization, is a mathematical method to achieve the minimum or maximum value of a linear function on a convex polyhedron. This convex polyhedron is, in fact, a graphical representation of some constraints as inequalities on/off functional variables. To put simply, we can achieve the best outcome (e.g. maximum profit or minimum cost) by using linear programming under specific settings and constraints. While linear programming is mainly used in management and economics, it can also be utilized for some engineering problems. (Hilier and Liberman, 2003). Background Linear Programming was developed as a mathematical pattern during World War II to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. The method was kept secret until After the war, many industries began using it. The founders of linear programming are: George Dantzig who published the Simplex method in 1947, John von Neumann who developed the theory of duality, and Leonid Kantorovich - the Russian mathematician who applied similar techniques before Dantzig and won the Noble Prize in Leonid Kantorovich showed, for the first time, in 1979 that the linear-programming problem was solvable in polynomial time, but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior-point method for solving linear-programming problems. (Hilier and Liberman,2003). Linear programming is continuously applied by the researchers to improve operations. The Babcock & Wilcox applied the linear programming to help plan a major expansion of the company s Tubular Products Division (TPD) in Pennsylvania (Drayer & Seabury, 1975). Own has also used the linear programming method to design antenna array patterns that suppress interference from certain directions (Owen & Mason, 1984). Expressing the problem Linear programming is a set of techniques and methods inferred from mathematics and other sciences which can play an efficient role in improving the management decisions. Although it is still regarded as a new science, but it has well proved to be capable in solving problems such as production planning, allocating resources, inventory control, and advertising. Those managers who care about the best outcomes for their decisions cannot be indifferent to this. Bourton, Gidley, Baker, and Reda-Wilson used linear programming in a study to choose the appropriate marketing strategy. In their study, optimized solutions as well as rather optimized solutions were determined and evaluated. (Mehdipoor et al, 2006) COPY RIGHT 2013 Institute of Interdisciplinary Business Research 701

2 The linear programming model is used by the managers to determine the most economical arrangement of finance, to arrange the best times to start and finish projects, and to select projects to minimize the total net present cost of capital (Wijeratne & Harris, 1984). The linear programming is a utility to select the desired pattern from among a variety of production plans (Edward et al, 2011). Linear programming optimizes (maximizing or minimizing) a dependant variable subject to a set of independent variables in a linear relationship, given a number of linear constraints of independent variables. The value of dependent variables which is the value obtained from solving the problem, is subject to the independent variables set by the decision maker (or determined by solving the problem). The dependent variables are usually set as objective function which may be one of the economic concepts such as profit, cost, income, production, sales, distance and time, etc. The independent variables in linear programming are known as variables of unknown value, and the decision maker has to calculate the value of such variables by solving the problem (Mehdipoor et al, 2006). Modeling The successful utilization of linear programming in various fields expanded the scientific use of this technique. There are resource constraints such as labor, time, space, and technology in every field which have to be optimized. Formulating a linear programming problem involves optimization elements such as profit or income, or minimizing elements such as cost, time, and distance. When the problem is expressed, the management objective is set, and the capability of the linear programming technique for the given problem is ensured, the next step in a real problemsolving situation is to express the problem in a mathematical model. (Guilani-nia, 2005) The elements of a linear programming model are: decision variables, objective function, and model constraints. The objective function and constraints of a linear programming model are the decision variables and parameters respectively. The decision variables include mathematical symbols which represent the level of activity of any organization. The objective function is a mathematical linear relationship which expresses the objective of the organization in terms of decision variables. The objective function is always set as maximizing or minimizing. The model constraints also express the linear relation among the decision variables. The constraints are imposed on the organization by the operational environment, and are often due to limited resources or the organization s internal policies (Azar, 1999)) When one designs the real decision constraint model, he/she may begin with a simple model which covers a part of the problem. Later on, more real constraints can be added during the next steps. Those with the impression that the whole model could be designed in one step would encounter a very complicated difficult task (Orlin, 2007). The linear programming techniques can be used only when they embody the required assumptions: proportionality, additively, divisibility, and certainty (Mehregan, 1993) Here are the prerequisites to a linear programming model: 1- Set and define the solution variables 2- Determine the linear relationship of solution variables 3- Express the constraints as a linear relationship of solution variables which represents the problem resources constraints 4- Availability of a mathematical relation within the variables 5- Non-negativity of the variables; in linear programming, the value of the variables after solving the problem must be positive, equal or more than zero (Guilani-nia, 2005) The linear programming models are presented in various forms of maximizing or minimizing the objective function and limits. There are two presentation forms for solving problems of linear programming: Canonical and Standard. The standard form is directly applied to solve the model. The canonical form is particularly useful in presenting the theory of duality. COPY RIGHT 2013 Institute of Interdisciplinary Business Research 702

3 Canonical form: Maximize X 0 = X j 0 i = 1, 2,..., m j = 1, 2,..., n In this form of linear programming, all decision variables are negative, all limits as ) ( and objective functions are maximized. Standard form: A 1 x 1 +a 2 x 2 s 1 = b s 1 0 P 1 x 1 +p 2 x 2 +s 2 = q s 2 0 The standard form the following characteristics: 1- All limits are as equality except non-negativity limit which is as inequality (0 ) 2- The L non-negative for each limit 3- All variables are non-negative 4- Objective function is either minimized or maximized (Taha, 1996) One of the primary and most important uses of linear programming is solving transportation problems. The earliest application of this method was experienced during World War II. Therefore, most books and publications on applied linear programming make references to network patterns in solving transportation problems (Afandizadeh,2003). A linear programming problem has two main characteristics: 1- A determined objective 2- Constraints to be satisfied (Mehdipoor et al, 2006) Linear programming method is applicable to problems where the objective function and are linear and all variables are non-negative (Saadeghi and Doosti, 2010) Linear programming is a mathematical technique which can be widely used in management planning provided there is a defined objective to be maximized or minimized. It involves a number of constraints to be satisfied, and the objective equalities and the constraints inequalities need to be shown in linear relationships (Hadawy, 2007). Solving the LP problem by Graphical Method: To solve a linear programming problem for each product (or variable), two axes (dimensions) are required. Therefore, a graph for a 2-dimensional model is easy. But for solving linear programming problems with more dimensions (variables), we have to use Simplex method and often computer software. Then, it would be better to solve such problems by graphical method. Although drawing lines for model constraints limits the region of possible solutions, but still there are many more solutions left. The final optimal value (solution) for the linear programming model occurs at one of the vertices of the region determined by the constraints. Combinations of x 1 and x 2 which lie in one of these vertices are called the basic solution. There are only two more steps left for a graphical method as following: 1- Find the vertices of the region. 2- Test the objective function at each of the vertices (profit obtained from selling a product). The vertex that has the maximum value is the optimal solution. Linear programming problems with 2 variables (dimensions) can be easily solved graphically, but solving more complicated models (with 3 or more variables) could be done by Simplex method (manually or using a computer). The table in which all figures on the first line are negative, contains the final optimized solution. In addition to the final solution, other useful information such as shadow prices will be provided in the final Simplex table. Also, the COPY RIGHT 2013 Institute of Interdisciplinary Business Research 703

4 allowed range of profit coefficient and limited capacities can be computed using the data in Simplex tables. The allowed range means that if the coefficients are altered within the defined range, the final solution will not change in terms of various combinations. Therefore, it may be said that estimating the coefficients which have a rather wide allowed range, is less risky. The shadow prices represent the value of objective function (profit or cost) for any increased unit of each of the limited resources (Hadawy, 2007). The Simplex algorithm developed by Dantzig solves LP problems by constructing a feasible solution at a vertex of the polytypic and then walking along a path on the edges of the polytypic to vertices with non-decreasing values of the objective function until an optimum is reached. Although the Simplex algorithm is, in practice, quite efficient and can find the global optimum if certain precautions against cycling are taken. In some cases, it has poor behavior known as worst-case. Some linear programming problems could be constructed for which the Simplex method takes a number of steps exponential in the problem size. For some time, it was not known whether the LP problem was solvable in polynomial time. Finally, this problem was solved by Leonid Khachiyan in 1979 by introducing the Ellipsoid method. This method was polynomial time. The algorithm of Khachiyan was not practically a break-through, as the simple method was more efficient for all but specially constructed linear programs. However, the theoretical aspect of Kachiyan's algorithm was of landmark importance. Khachiyan's algorithm inspired new lines of research to solve the linear programs called Interior-point method. The interior-point methods move through the interior of the feasible region towards the optimal point (Hilier and Lieberman, 2003). Conclusion In this paper, linear programming and its applications have been introduced, and one of the methods to solve LP problems has been completely explained. Linear programming is a skill/method involving mathematical techniques which can help many careers from management to engineering find the optimal solution. Given the limited resources in the ناآرام environment of today communities, using linear programming is organizations is an efficient solution. Knowing this skill and its characteristics could be a triumphant solution critical situations. COPY RIGHT 2013 Institute of Interdisciplinary Business Research 704

5 References 1- Afandizadeh, Sh & Doosti, R. (2003). Application of linear programming to improve transportation of wheat countrywide, the 6th International Conference of Civil Engineering. 2- Azar, A. (1999). Operations Research I. Payam-e Noor Publications. 3- Drayer,W & Seabury,S. (1975). "Linear programming A case example", Strategy & Leadership, Vol. 3 Iss: 5, pp Edwards. D. J & Malekzadeh.H & Silas B. Y. (2001). "A linear programming decision tool for selecting the optimum excavator", Structural Survey, Vol. 19 Iss: 2, pp Guilani-nia, Sh. (2005). Advanced Research in Operations (Applied Concepts). 6- Hadawy, A. (2006). Production Budgeting and Linear Programming, The Islamic Azad University of Boroojerd 7- Hamdi, Taha. (1996). An Introduction to Operations Research translated by Mohammad Bagher Bazargan. Tehran University s Publication Center 8- Hilier. F. S & Lieberman,J.D. (2003). Research in Operation translated by Mohammad Modarres and Ardawan Asef Vaziri, 10th printing in Tehran by Nashr-e Javan publications 9- Mehdipoor, E. & Sadr-ol-ashraafi, S. M. & Karbaasi, A. (2006) A Comparison of Canonical Linear Programming Techniques, Meaty Chicken Feed Framing With Linear Programming Models, Scientific- Research Magazine of Agriculture, 12th year, issue no Mehregan, M.R. (1993). Operational Research. Saalekaan Publications 11- Orlin,J. B.(2007). Optimization Methods in Management translated by Mohammad-Reza Hamidizadeh 12- Owen. P & MASON, J.C. (1984). "The use of linear programming the design of antenna pattern with prescribed nulls and other constraints", compel: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 3 Iss: 4, pp Saadeghi, Gh & Doosti, R. (2007).A review on published articles about various methods of optimized product distribution in Wijeratne. N.N, F.C. Harris. (1984). "Capital Budgeting Using a Linear Programming Model", International Journal of Operations & Production Management, Vol. 4 Iss: 2, pp COPY RIGHT 2013 Institute of Interdisciplinary Business Research 705