**FUNDAMENTALS OF SENSITIVITY AND PARAMETRIC ANALYSIS**

**OF A LINEAR PROGRAMMING PROBLEM**

**TABLE OF CONTENT**

Title Page- – – – – – – – -i

Dedication – – – – – – – -iii

Table of Content – – – – – – -v

Abstract – – – – – – – -vii

**CHAPTER ONE**

1.0 Introduction – – – – – – -1

1.1 Background of the Study – – – – -2

1.2 Parametric Analysis – – – – – -5

1.3 Statement of Problem – – – – – 6

1.4 Significant of the Study- – – – – -7

**CHAPTER TWO**

2.0 Literature Review – – – – – – – -8

2.1 Classification of Sensitivity and Parametric Analysis- – -14

**CHAPTER THREE**

3.0 Parametric Analysis- – – – – – – -16

3.1 The LP Model and Continuous Changes in the R.H.S.- – -18

3.2 Background of the Procedure- – – – – -19

3.3 The Algorithm – – – – – – – -22

**CHAPTER FOUR**

4.0 Numerical Illustration and Discussion – – – – -33

4.1 Numerical Illustration of Parametric- – – – -33

4.2 Discussion and Analysis- – – – – – -47

**CHAPTER FIVE**

** SUMMARY AND CONCLUSION**

5.1 Summary- – – – – – – – -49

5.2 Conclusion- – – – – – – – -55

** **References – – – – – – – – -57

** **

** **

**CHAPTER ONE**

**1.0 INTRODUCTION**

Sensitivity analysis is the study of what happens to the optimal solution when discrete changes occur in the original coefficients of an LP problem. The changes in this case may be in the form of additional constraints or variables.

A Mathematical Model Comprises of independent and dependents variables and a system of relationship in the form of equations or inequalities that exist between the variables. Mathematically, the numerical methods employed to solve the equations underlying the mathematical model are often the important aspect of the model development process. Furthermore, mathematical models also include variable parameters which constitute the relationship that unfold from experiments and as such, their actual values are not precisely known precisely but may vary within some ranges of uncertainty. The sensitivity analysis of a mathematical model becomes even more complicated if the numerical calculations cannot be handled with known methods.

The simplest and also the most common procedure for assessing the effects of parameter variations on a model’s result is to vary selected input parameters, Rerun code and record the corresponding changes in the result. The model parameters responsible for the largest relative changes are classified to be the most important. For complex models, though the large amount of computing time needed by such re-calculations severely restricts the scope of this sensitivity analysis. In practice, this means that the modeler can investigate only a few parameters that he judges a priori to be important. Cacuci (2002).

**1.1 BACKGROUND OF THE STUDY**

There are certain questions that are often asked regarding optimal solution of a linear programming (LP) problem. For example, what happens to the optimum solution (both the values of the variable and the value of the objective function) when certain changes occur in some of the values of the original data? These are some of the questions that sensitivity analysis tries to answer. If the objective function or the variables change when an original coefficient is changed then we say that the optimal solution of the programming is sensitive. A sort of examination of the impact of the input data on the output result is crucial. The procedures and algorithms of mathematical programming are important but the problems that one often encounter in practice are usually associated with data; getting it at all and getting accurate data. Some data, necessary for mathematical models are inherently uncertain. Consider profit per item, for example, which is approximated from estimates of the fluctuating costs of raw materials, expected sales volumes, labour costs etc what you want to know from sensitivity analysis is which data has a significant impact on the results: then you can concentrate on getting accurate data for those items or at least running through several scenarios with various values of the crucial data in place to get an idea of the range of possible outcomes, there are several ways to approach sensitivity analysis. If your model is small enough to solve quite quickly, you can resolve manually (though computationally) by simply changing the initial data.

This can be done as many times as needed. On the other hand, if the model is very large and takes a long time to solve you can apply on the relationship between the initial tableau and only later tableau (in particular the optimum tableau to quickly update the optimum solution when changes are made to the coefficients of the original tableau Taylor (1986).

Uncertainty in a model can have different origins in different decision problem. It may be due to either incomplete information or fluctuations inherent in the problem or unpredictable changes in future. Unlike the ordinary sensitivity analysis the construction presented here allows us for any dependent, independent, multiple changes in both the right-hand side (RHS) values and the cost coefficients of linear program models having unique solution. This is one way of tackling uncertainties in a more “deterministic” manner. The approach is at times called by various names such as “Scenario Modeling”, ‘deterministic modeling”, sensitivity analysis” and “stability analysis, the idea is to subjectively come up with a ranked list of higher level uncertainties that might presumably have a bigger impact on the eventual mapping result. An understanding of the influence of the above on the course of action suggested by the model is cardinal because:

- Different level of acceptance (by the general public, the decision makers, stakeholders) may be attached to different types of uncertainty.
- Differently on the reliability robustness and efficiency of the construct. Example of settings where the above applies are many and include: construction of economic and environmental information.

**1.2 PARAMETRIC ANALYSIS**

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