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In order to survive and lead today’s highly competitive and demand driven market, there is a great deal of pressure on management to make economical decisions. One of the essential managerial skills is the ability to allocate and utilize resources appropriately in the efforts of achieving the optimal performance efficiently. Mathematical models have long played an important role in management and economics but it is only in the last few decades that management per se has received the sort of rigorous study that permits the application of mathematics. It is one of the primary tools used by operations researchers in making decisions.
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Mathematical programming is the OR technique that has been most widely applied in management and economics. It has been used to solve a considerable range of problems in financial markets – forming portfolios of equities, bonds, loans and currencies, generalized hedging, capital budgeting, cash management, insurance management, equity and bond index tracking, estimating the implied risk neutral probabilities for options, designing leveraged leases, computing the maximum loss sustained by shareholders, spotting insolvent banks, sorting out the failure of a stock exchange and understanding the forces leading to financial innovations. It deals with models comprising of an objective function and a set of constraints.
An objective function is a mathematical expression of the quantity to be maximized or minimized. For example, manufacturers may wish to maximize production or minimize costs, advertisers may wish to maximize a product’s exposure, and financial analysts may wish to maximize rate of return.
Constraints are mathematical expressions of restrictions that are placed on potential values of the objective function. For example, production may be constrained by the total amount of labor at hand and machine production capacity, an advertiser may be constrained by an advertising budget, and an investment portfolio may be restricted by the allowable risk.
Symbolically, mathematical models in operations research may be viewed generally as determining the values of the decision variables which will
The function is the objective function while represents the constraint, whereis constant. The constraintsare the non-negativity constraints. In general,optimization of the objective function signifies either amaximization or minimizationof this function.
The types of the objective and constraint functions of a mathematical model depend directly on the systems which they represent. Thus, these functions may be linear or non-linear. Also the decision variables may be continuous or discrete and the parameters of the system may be deterministic or probabilistic.
The result of this diversity in system representation is the development of a corresponding number of optimization techniques suitable for solving these models. These mainly include linear programming, integer programming, goal programming, non-linear programming, dynamic programming, and stochastic programming.
The Mathematical Programming Approach
There is an orderly sequence of steps that is followed for a systematic formulation, solution and implementation of a mathematical programming model. These steps could be applied to the development of any management science model.
Although the practical applications of mathematical programming cover a broad range of problems, it is possible to distinguish five general stages that the solution of any mathematical programming should follow. These stages along with their main characteristics are as follows :
A. Formulating the Model
The first step to be taken in a practical application is the development of the model. The following are the elements that define the model structure :
- Selection of a time horizon
- Selection of decision variables and parameters
- Definition of constraints
- Selection of the objective function
B. Gathering the Data
After defining the model, the data required to define the parameters of the model must be collected. This involves the data regarding the objective function coefficients, the constraint coefficients and the right hand side of the constraints.
C. Obtaining an Optimal Solution
Due to the lengthy nature of calculations required to obtain the optimal solution of a mathematical programming model, a digital computer is invariably used in this stage of model implementation. Nowadays, all the computer manufacturers offer highly efficient codes to solve mathematical programming models.
D. Applying Sensitivity Analysis
Sensitivity analysis, also called post-optimum analysis, is performed on the optimal solutions obtained for the linear programming problems originally formulated. This analysis is important for several reasons :
- Data uncertainity
- Dynamic considerations
- Input errors
E. Testing and Implementing the Solution
Once the optimal solution is obtained, it should be tested fully to ensure that the model clearly represents the real situation. The importance of conducting sensitivity analysis as part of this testing effort has already been discussed. If the solution is unacceptable, new refinements have to be incorporated in the model and new solutions obtained until the mathematical programming model is adequate.
When the testing is over, the model can be implemented. Implementation usually means solving the model with real data to reach a decision or a set of decisions.
Types of Mathematical Programming
A. Linear programming
Linear programming is one of the most successful disciplines within the field of Operational Research. It arose as a mathematical model developed during the second world war to plan expenditures and returns in order to reduce costs to the army and increase losses to the enemy. It was kept secret until 1947. Post war, many industries found its use in their daily planning.
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Linear programming is a technique for making decisions under certainity i.e.; when all the courses of options available to an organization are known in advance and the objective of the firm along with its constraints are quantified. Out of all the possible alternative that course of action is chosen that which yields the optimal results.
Mathematically, linear programming problem calls for finding non-negativeso as to maximize a linear functionsubject to a system of linear equations :
There are two extensions of Linear programming which are as follows :
a) Integer Linear programming
The Integer Linear Programming (ILP) is an extension of Linear programming where all of the variables must only take on integer values.
b) Mixed Integer Linear Programming
The Mixed integer Linear Programming (MILP) is also an extension of the Linear programming, but some of the variables must take on integer values while others take on real values.
B. Quadratic programming
Quadratic Programming is a special type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a quadratic function of several variables subject to linear constraints on these variables.
C. Goal Prgramming
Goal Programming is promulgated as an aid for decision-making problems with multiple, possibly conflicting goals. In a typical goal programming model, each goal is formulated as a constraint. There are two variables associated with each goal (each constraint), over-achievement deviation and under-achievement deviation. The value of these two deviational variables measure how well the corresponding goal is accomplished.
An objective function in a goal programming formulation is usually a linear function in deviational variables. Specifically, the objective function takes the weighted summation of the deviational variables. The weights assigned to a deviational variable indicate the importance of the corresponding goal in decision-making process. The objective is thus to minimize the weighted sum of deviations from goal achievement, i.e., to accomplish the best overall achievement.
D. Dynamic Programming
Dynamic Programming is a mathematical programming technique which fragments a large problem into several smaller problems. The approach is to solve the all the smaller, easier problems individually in order to reach a solution to the original problem.
This technique is useful for making decisions that consist of several steps, each of which also requires a decision. In addition, it is assumed that the smaller problems are not independent of one another given they contribute to the larger question.
E. Stochastic Programming
Many optimization problems are described by uncertain parameters. When these uncertain parameters can be considered as random variables and have known probability distributions, new optimization problems can be formulated that involve expected values of these random variables. In this manner, a new problem (called the deterministic equivalent) is obtained that removes the uncertainty and results in a deterministic optimization problem. This is the approach of Stochastic Programming.
Application areas of Mathematical Modeling
The applications of mathematical modeling in management and economics today are so manifold that it is difficult to list them down. A few of the major application areas of mathematical models are as follows :
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