# Management Science And Linear Programming

1990 words (8 pages) Essay

17th May 2017 Management Reference this

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Management science is the application of a scientific approach to solving management problems in order to help managers make better decisions.”1 In this essay, I am going to describe the way in which optimization techniques contributing to the management decision making progress, and address the advantages and disadvantages of using these techniques and provide some examples of applications of linear programming to illustrating my discussion.

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Management Science can be used in a variety of organizations to solve plenty of problems. The applications of management science techniques are widespread, and they have been credited with increasing the efficiency and productivity of business firms frequently. According to various surveys of business, many indicate that they use management science techniques, and most are satisfied with the results. Typically for management science, the problem is studied when we build a model for it, such a model is often a mathematical model. It can be in the form of chart or graph, but most frequently a management science model consists of a set of mathematical relationships. These relationships are made up of numbers and symbols. Once model is built, algorithms are used to solve problem. Various techniques are devised to model problem and solve it for possible solutions.

Linear programming is one of the widely used modeling techniques that can solve decision problems with many thousands of variables. Linear programming models include an objective function and model constraints, which consist of decision variables and parameters. The objective function is a linear mathematical relationship that describes the objective of the firm in terms of the decision variables. The objective function always consists of a certain number of variables, (e.g., maximize the profit or minimize the cost of producing radios). The model constraints are also linear relationships of the decision variables used in the objective function. This technique is closely related to linear algebra and uses inequalities in the problem statement rather than equalities. A linear programming problem can fall in three categories: problems with more than one optimal solution, infeasible problems, and problems with unbounded solutions. In an optimal solution, the objective function has a unique maximum or minimum value. An infeasible problem has no feasible solution area; every possible solution point violates one or more constrain. A problem is unbounded if the constraints do not sufficiently restrain the objective function so that for any given feasible solution, another feasible solution can be found that makes further improvement to the objective function. Linear programming problems can be solved using graphical analysis method.

The mathematical-optimization methods of linear programming and quadratic programming have both been used in solving nutrition problems. Quadratic programming is a special type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a quadratic function of several variable subject to linear constraints on these variables.2 Although quadratic programming is much less widely used than linear programming. In nutrient estimation, quadratic programming differs from linear programming in that it more heavily penalizes large nutrient differences than does linear programming. To estimate the unknown nutrient values in food products, the mathematical model minimizes the differences between calculates nutrient values and the known nutrient values of a food product. For linear programming it minimizes the sum of the absolute values of the differences, while for quadratic programming it minimizes the sum of the squares of the differences. Nutritionists now use a production version of the software daily as decision support tool for maintaining food-composition databases. Usually, the nutritionist employs linear programming to derive an initial estimate of ingredient amounts as needed by manually overriding the linear programming estimated amounts. This allows nutritionists to quickly obtain feasible estimated ingredient amounts, which they can further refine using their knowledge of product formulation and food composition. Because quadratic programming has no apparent advantages and is slower than linear programming, quadratic programming is not in use. For simplicity, we use a constant weight tolerance even though the difficulty of obtaining a feasible solution varies. As of April 1996, nine nutritionists were using the software, five were using it weekly, and three on a daily basis. Mathematical optimization has increased the efficiency with which food-nutrient values are estimated, even though nutrient tolerances were made more stringent than in trial- and error methods.3

Probabilistic techniques are another class of modeling approach for problem solving. These techniques are distinguished from mathematical programming techniques in that the results are probabilistic. In this technique risk means uncertainty for which the probability of distribution is know. Therefore risk assessment involves study of the outcomes of decisions along with their probabilities. Probability assessment tries to fill gap between what is know and what need to be know for an optimal solution. Therefore, probabilistic models are used to prevent events happening because of the adverse uncertainty. Decision analysis and queuing systems are example of probabilistic techniques. The modeling techniques use to solve physical problems such as transportation or flow of commodities is Network modeling. Network problems are an abstract representation of processes and activities for a given problem and illustrated by using network branches and nodes. This technique uses most cost effective way to transport the goods, and to determine maximum or minimum possible flow from source to destination and to find shortest critical path in large projects. 4

By means of one example I am going to gradually check the advantages and disadvantages of linear programming. First of all it is known that one of the main advantages of linear programming is that it fits strictly with reality, as I will see, the example reflects this property. Suppose we are running a football club and launching a new merchandising campaign and we have to decide the quantity of scarves and shirts produced, considering current constraints. The sale prices of shirts and scarves are £35 and £10 respectively, also we know the maximum annual manufacture capacity is 2000 units, secondly four times more time is needed to sew a shirt compared with a scarf having at the most 2300 hours a year and finally space is limited up to 2500 square inches, requiring shirts and scarves, 3 and 2 square inches respectively.5 The first advantage is the calculation facility, as can be checked in the first step where we have to model or formulate the problem. This is a process where verbal statement is translated to mathematical statement. The incomes must be maximized knowing the different prices of scarves and shirts but some limitations have been set which are called constraints, in this case limitations are related with capacity, time and sales space.

Objective function: 35X+10Y

Constraints:

1. capacity x+yâ‰¤2000

2. time 4x+yâ‰¤2300

3. sales space 3x+2yâ‰¤2500

4. nonnegativity constraints x, yâ‰¥0

The maximum profit is in the point where the second and third constraints intersect each other. As a result is known that X=420 and Y=620 .

To draw the objective function to make the maximum profit equal to 3000, but it means nothing, because we can choose any number, the slope of the line is the matter that concerns us. If we move parallel the function toward larger objective function values, the maximum profit point will be found when the line will be completely outside of the feasible region. It can be ascertained that in frequent occasions some variables are ignored in this sort of problems. Hence the problem is less rigorous and it loses accuracy and certainly, that becomes a disadvantage. This example has been resolved by graphical method because it has only two variables. It is impossible to solve problems with the graphical method if there are more than two variables. Therefore this is another disadvantage of linear programming: graphical method can be used only under determined conditions. But there still are more advantages, linear programming analysis can help both with determining whether management’s plans are feasible and in unbounded cases where the value of the solution is infinitely large, without violating any of the constraints, warning us that the problem is improperly formulated. The next step to analyze is what happens when we change the values of the objective function or in the constraints. This is another advantage of using linear programming, when can check easily how the results vary if we change the old coefficients for others. This is called sensitivity analysis, which determines how changes affect the optimal solution to the original linear programming problem.

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Linear programming is the most used program in the management area and there are several reasons which take us to select this method solving management problems owing to the complexity of the problems that can be handled.

The management science modeling process helps businesses to improve their operations through the use of scientific methods and the development of specialized techniques. It is the process of re searching for an optimal solution to the existing problem. Management science modeling process provides systematic, analytical and general approaches to the problem solving for decision-making, regardless of the nature of the system, product, or service. Models are aimed at assisting the decision-maker in decision-making process.

Management science is the application of a scientific approach to solving management problems in order to help managers make better decisions.”1 In this essay, I am going to describe the way in which optimization techniques contributing to the management decision making progress, and address the advantages and disadvantages of using these techniques and provide some examples of applications of linear programming to illustrating my discussion.

Management Science can be used in a variety of organizations to solve plenty of problems. The applications of management science techniques are widespread, and they have been credited with increasing the efficiency and productivity of business firms frequently. According to various surveys of business, many indicate that they use management science techniques, and most are satisfied with the results. Typically for management science, the problem is studied when we build a model for it, such a model is often a mathematical model. It can be in the form of chart or graph, but most frequently a management science model consists of a set of mathematical relationships. These relationships are made up of numbers and symbols. Once model is built, algorithms are used to solve problem. Various techniques are devised to model problem and solve it for possible solutions.

Linear programming is one of the widely used modeling techniques that can solve decision problems with many thousands of variables. Linear programming models include an objective function and model constraints, which consist of decision variables and parameters. The objective function is a linear mathematical relationship that describes the objective of the firm in terms of the decision variables. The objective function always consists of a certain number of variables, (e.g., maximize the profit or minimize the cost of producing radios). The model constraints are also linear relationships of the decision variables used in the objective function. This technique is closely related to linear algebra and uses inequalities in the problem statement rather than equalities. A linear programming problem can fall in three categories: problems with more than one optimal solution, infeasible problems, and problems with unbounded solutions. In an optimal solution, the objective function has a unique maximum or minimum value. An infeasible problem has no feasible solution area; every possible solution point violates one or more constrain. A problem is unbounded if the constraints do not sufficiently restrain the objective function so that for any given feasible solution, another feasible solution can be found that makes further improvement to the objective function. Linear programming problems can be solved using graphical analysis method.

The mathematical-optimization methods of linear programming and quadratic programming have both been used in solving nutrition problems. Quadratic programming is a special type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a quadratic function of several variable subject to linear constraints on these variables.2 Although quadratic programming is much less widely used than linear programming. In nutrient estimation, quadratic programming differs from linear programming in that it more heavily penalizes large nutrient differences than does linear programming. To estimate the unknown nutrient values in food products, the mathematical model minimizes the differences between calculates nutrient values and the known nutrient values of a food product. For linear programming it minimizes the sum of the absolute values of the differences, while for quadratic programming it minimizes the sum of the squares of the differences. Nutritionists now use a production version of the software daily as decision support tool for maintaining food-composition databases. Usually, the nutritionist employs linear programming to derive an initial estimate of ingredient amounts as needed by manually overriding the linear programming estimated amounts. This allows nutritionists to quickly obtain feasible estimated ingredient amounts, which they can further refine using their knowledge of product formulation and food composition. Because quadratic programming has no apparent advantages and is slower than linear programming, quadratic programming is not in use. For simplicity, we use a constant weight tolerance even though the difficulty of obtaining a feasible solution varies. As of April 1996, nine nutritionists were using the software, five were using it weekly, and three on a daily basis. Mathematical optimization has increased the efficiency with which food-nutrient values are estimated, even though nutrient tolerances were made more stringent than in trial- and error methods.3

Probabilistic techniques are another class of modeling approach for problem solving. These techniques are distinguished from mathematical programming techniques in that the results are probabilistic. In this technique risk means uncertainty for which the probability of distribution is know. Therefore risk assessment involves study of the outcomes of decisions along with their probabilities. Probability assessment tries to fill gap between what is know and what need to be know for an optimal solution. Therefore, probabilistic models are used to prevent events happening because of the adverse uncertainty. Decision analysis and queuing systems are example of probabilistic techniques. The modeling techniques use to solve physical problems such as transportation or flow of commodities is Network modeling. Network problems are an abstract representation of processes and activities for a given problem and illustrated by using network branches and nodes. This technique uses most cost effective way to transport the goods, and to determine maximum or minimum possible flow from source to destination and to find shortest critical path in large projects. 4

By means of one example I am going to gradually check the advantages and disadvantages of linear programming. First of all it is known that one of the main advantages of linear programming is that it fits strictly with reality, as I will see, the example reflects this property. Suppose we are running a football club and launching a new merchandising campaign and we have to decide the quantity of scarves and shirts produced, considering current constraints. The sale prices of shirts and scarves are £35 and £10 respectively, also we know the maximum annual manufacture capacity is 2000 units, secondly four times more time is needed to sew a shirt compared with a scarf having at the most 2300 hours a year and finally space is limited up to 2500 square inches, requiring shirts and scarves, 3 and 2 square inches respectively.5 The first advantage is the calculation facility, as can be checked in the first step where we have to model or formulate the problem. This is a process where verbal statement is translated to mathematical statement. The incomes must be maximized knowing the different prices of scarves and shirts but some limitations have been set which are called constraints, in this case limitations are related with capacity, time and sales space.

Objective function: 35X+10Y

Constraints:

1. capacity x+yâ‰¤2000

2. time 4x+yâ‰¤2300

3. sales space 3x+2yâ‰¤2500

4. nonnegativity constraints x, yâ‰¥0

The maximum profit is in the point where the second and third constraints intersect each other. As a result is known that X=420 and Y=620 .

To draw the objective function to make the maximum profit equal to 3000, but it means nothing, because we can choose any number, the slope of the line is the matter that concerns us. If we move parallel the function toward larger objective function values, the maximum profit point will be found when the line will be completely outside of the feasible region. It can be ascertained that in frequent occasions some variables are ignored in this sort of problems. Hence the problem is less rigorous and it loses accuracy and certainly, that becomes a disadvantage. This example has been resolved by graphical method because it has only two variables. It is impossible to solve problems with the graphical method if there are more than two variables. Therefore this is another disadvantage of linear programming: graphical method can be used only under determined conditions. But there still are more advantages, linear programming analysis can help both with determining whether management’s plans are feasible and in unbounded cases where the value of the solution is infinitely large, without violating any of the constraints, warning us that the problem is improperly formulated. The next step to analyze is what happens when we change the values of the objective function or in the constraints. This is another advantage of using linear programming, when can check easily how the results vary if we change the old coefficients for others. This is called sensitivity analysis, which determines how changes affect the optimal solution to the original linear programming problem.

Linear programming is the most used program in the management area and there are several reasons which take us to select this method solving management problems owing to the complexity of the problems that can be handled.

The management science modeling process helps businesses to improve their operations through the use of scientific methods and the development of specialized techniques. It is the process of re searching for an optimal solution to the existing problem. Management science modeling process provides systematic, analytical and general approaches to the problem solving for decision-making, regardless of the nature of the system, product, or service. Models are aimed at assisting the decision-maker in decision-making process.

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