Linear Programming

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Exercise 1.

Sensitivity analysis is the understanding of the manner in which uncertainty in the output of a mathematical replica can be shared with other sources of uncertainty in its input. This process is usually important because it develops a better comprehension of the relationship between outputs and inputs within a mathematical model. Similarly, it helps to test the strength of the results of a mathematical replica when a form of uncertainty exists. Therefore, it is a useful method of identifying any errors within a system as a result of the relationship between the input and output.

Exercise 2.

The linear programming model has algebraic linear expressions that offer a description of the constraints and the objective function. This model is useful in maintaining the level of profitability in a company by maximizing the resources and minimizing the costs. Thus, this system is practical in the majority of business enterprises that seek to minimize the costs of their operations while satisfying the conservation requirements of flow at each node. Business enterprises rely on this system to make their business activities efficient and to maximize profits in the long run as a result of the careful observation of the input and output flows.
Thus, the nodes represent the critical business activities of the enterprise responsible for generating profits. At the end of each node, an arc springs and it represents the line segments within the business enterprise. Thus, the business makes use of the resources it has to efficient levels to reduce the costs of operation within the market. Therefore, the flow of the network follows a definite course, entering and leaving the nodes leading to an arc. Therefore, the flow entering each node must be equal or similar to those leaving it with the arcs representing the decision variables for the network flow.

Exercise 3.

A general property of linear programming solutions is that the integer variables represent the decisions of a business. These variables can only be 0 or 1, and this helps to identify the preciseness of a business decision. The integer value must not be fractional because it will not make sense for the application of the problem. For example, when a business is making decisions, it will conclude to make a definite number of items in the decision making process. Thus, it is not possible for the business to manufacture, for example 2.4 computers because this is not a rational solution.
Truncating or rounding off non-integer values of a business is not an appropriate method for obtaining integer solutions. The solution has to be precise in regard to the demands and objectives of the enterprise. Thus, the integer variables are direct representations of the quantities of the decision that the business has to do. Therefore, they can only be complete integer values for the decision of the business to make sense and be applicable. A decimal result is not a real reflection of the quantities that an enterprise has to consider in the decision making process.

Exercise 4.

A single channel waiting line consists of a single line shaping at the front of a single server. In this system, the service to the customer is on a first come, first serve basis where there is service to only one customer at a time, for example customers in a fast food restaurant. The multiple single channel waiting lines consist of single lines shaping before multiple servers. This system allows for the service of two or more customers at once and a good example is the banking halls where customers line up for service at different cashiers (Gershenfeld 69).
The multiple channels waiting line consist of a number of waiting lines with multiple servers all functioning at once. In this system, it is possible to serve each customer at once because of the availability of servers, for example the printing process of documents. The preferable waiting line is the multiple channel waiting line because there is a reduction in the time for attending to customers as each of them can access services at the same time. This system is also preferable because it utilizes the latest technology and thus increases the efficiency of the entire process.

Exercise 5.

A cyclical time series pattern happens in a revolving manner where the set of data or situation repeats itself. This means that there is a definite sequence of the data points, and the successive measurement happens over and over again. There are uniform time intervals that capture the set of events happening within this time series as they reach the end and go back to the initial starting point. The cyclical time series is a representative aspect of the basic laws of nature that enable a repetitive cycle in the sequence of events in a definite situation.
An excellent example of this phenomenon is the flow of a water body such as a river from its source to the exit point. A river originates from an uphill area where water and flows through a course to its final destination. However, a river pouring into a lake or sea is not the definitive end of the cycle mainly because this time series sequence repeats itself through the evaporation process. Thus, the water flowing into a larger water body will evaporate, form clouds and return to its origin through rainfall and begin the process all over again.

Exercise 6.

Simulation is the process of imitating a real world system or process through a definite period of time. There has to be the creation of a model that represents the actual situation in order to derive accurate imitations. The model is a direct representation of the processes and systems within the real world simulation over a period of time. Some of the most appropriate situations by employing simulation practices include the scientific testing of different experiments, the training process as well as the development of video games. These models depict actual, real life situations in order to draw a conclusion about possible improvements.
The simulation process is a useful source of knowledge because it provides insight into the mechanisms of different aspects of life. In the case of video games, it provides a realistic form of entertainment where the user can immerse themselves into a virtual world. The key characteristic that a model for simulation must have is a representation of the abstract and the physical nature of the real world. Similarly, another key characteristic of the model is that it must be smaller and hence cheaper to obtain to enable the study of the characteristics of the actual object. Thus, the model must be able to develop a synthetic environment of the real life situation.

Work Cited

Gershenfeld, Neil. The Nature of Mathematical Modeling. Cambridge: Cambridge University Press, 2000. Print