## Introduction

Linear programming can be referred to as a mathematical means for deciding the outcome of a function such as maximum or minimum as per relationships known as linear relationships, which are presented in a mathematical model (Hillier et al. 2010). It can also be referred to as a method for determining linear type functions in relations to their equality or inequality constraints.

On the other hand, non linear programming refers to a process by which a set of equalities and inequalities are collectively solved over a set of known variables in comparison to a function which is either maximized or minimized provided that the function or some of the set of variables are nonlinear (Strayer, 1989).

A nonlinear equation curves at some point or at many points as per the complexity of your equation. Also, non linear equations consists of exponents and the higher the exponents the more the curves when they are graphed. However, in a linear equation, the exponents are not higher than one and thus linear equations are on a straight line and their purpose is to find the line that comes closest to your data (Hiller & Hiller, 2010).

Sometimes, due to the close relationship between linear and nonlinear programming, it is difficult for managers to decide which of the two to use in decision analysis, forecasting and queuing in quantitative decision making.

However, nonlinear programming often provides a greater precision for problem solving to managers though it is more difficult to calculate due the exponent compositions. Linear programming can be used by managers for preliminary analysis whereas nonlinear programming, due to its precision, is best for the final analysis (Hiller & Hiller, 2010)

It is difficult to tell the difference between a linear programming and a nonlinear programming in almost every aspect. This is because both of them share the fact that decisions are made regarding the levels of a number of activities that have any value that satisfy given set of constraints. Also, decisions in reference to these activities are based on overall measure performance. The applications of both differ in the following three ways;

- The first difference is found in the nature of their relationships. It is a proven fact that nonlinear programming is used to model non proportional relationships between activity levels and overall measure of performance while linear programming assumes a proportional relationship.
- The second major difference is that while the constructing nonlinear formulas used in nonlinear programming, it posses a greater challenge than constructing linear formulas.
- The final major difference lies in the solution of linear and nonlinear equations. It is difficult to solve a nonlinear equations and sometimes impossible unlike the linear equations which are easy and straightforward.

Thus, it can be concluded that nonlinear programming raises various challenges because it uses a more complicated relationship between the activity levels and the overall measure of the performance.

The four types of profit graphs with Non proportional Relationships are;

### Decreasing marginal returns graph

When a graph is plotted such that profit is against the level of activity and the slope of the graph doe not in any way show increment as activity levels increase, such a graph is known as a decreasing marginal returns graph. Also, an activity is said to have decreasing marginal returns when the slope of the cost graph increases as the level of activity increases. You can also encounter decreasing marginal returns when less efficient inputs are used to increase the activity levels (Strayer, 1989).

### Piecewise Linear

These are termed thus because they consist of sequences of connected line segments and the slope of the profit graph remains the same within each line segment as the level of activity increases. However, the slope of the profit graph decreases at the kink where the next line segment begins and this graph can also be termed as having decreasing marginal returns (Karlin, 1959).

### Discontinuities graph

This is a situation where the profit graph is disconnected because it suddenly jumps up or slows down due to various reasons such as when the quantity discounts for the purchase of a product is available at a time the production levels for the said product has risen beyond specific levels (Strayer, 1989).

### Increasing marginal returns graph

This is another way in which proportionality assumption is violated. In this case, the slope of the profit graph does not decrease but at times increases with increase in the level of activity.

The four types of decision criteria are as listed below:

### The maximax Criterion

This is the decision criterion for they who are always optimistic and it states that a manager should only look on the positive side of events. This works through the identification of the maximum payoff from each decision alternative and then finding the maximum of these payoffs before choosing the corresponding decision alternatives.

Where this criterion is preferred, it is because it gives an opportunity for the best possible outcome to happen (Karlin, 1959). However, its drawback is that it ignores all the payoffs except the largest one. This means that all the rest are ignored despite their probability of success.

### The Maximin Criterion

This is the criteria used by total pessimists because it focuses on the worst that can happen. It works through the identification of the minimum payoff from all decision alternatives and thereafter finding the maximum of these minimum payoffs before choosing a corresponding decision alternative.

The rationale behind it is that it provides the best possible protection against any bad luck despite the likelihood of the decision leading to its worst state. However, it has a drawback in that it completely ignores all probabilities (Strayer, 1989).

### The Maximum Likelihood Criterion

It focuses on the most likely state of nature by identifying that state of nature with the largest priority and choosing the decision alternative that has the largest payoff for the given state of nature.

### Bayes decision rule

This uses the prior probabilities of the possible states of nature by using the calculation of the weighted average of its payoff through multiplication of each payoff by the prior probability of the corresponding state of nature and then summing the products. This weighted average is termed as the expected payoff and according to Baye’s rule, the largest of this expected payoff is chosen (Karlin, 1959).

## Decision trees and the analysis process

A decision tree can be termed as the process under which analysis progresses and decisions are made as per the analysis undertaken. Trees consist of nodes and branches. Nodes are points in the tree where events occur whereas branches can be referred to as lines coming from the nodes.

There are two types of nodes; decision nodes represented by squares and indicating a point which a decision is made and an event node which is circular in nature and indicates the occurrence of a random event. Decision trees are useful for analysis and visualization of problems especially problems of complex nature. Thus, decision trees can be termed as critical in the analysis process (Gale, 1960).

The five types of Forecasting Techniques and the difference in each approach are:

### The last value forecasting method

It is also called the naïve method and uses the last month’s sales as the forecast for the next moth. It is a reasonable method in situations where conditions change fast that previous sales before the last month are not reliable indicator of future sales.

### The Averaging forecasting method

This is the use of the average value of all the monthly sales to date as the forecast for next month. It is used when conditions tend to remain stable throughout such that earlier sales are reliable indicator of future sales.

### The moving-average forecasting method

This provides a middle-ground between the last value and the averaging method of forecasting. It does this by using the average of the monthly sales for only the recent months as the forecast for the next month.

### The exponential smoothing forecasting method

This provides a more complex version of the moving-average method in that it gives consideration to sales in only the most recent months. It does this by giving the greatest weights to the last month and then progressively smaller weights to older months.

### The exponential smoothing with trend

This is a more sophisticated form of the exponential smoothing forecasting. It adjusts exponential forecasting by directly considering any current upward or downward trend in sales and uses a two dimensional graph with sales measured along the vertical axis and time measured along the horizontal axis.

## Mean Absolute Deviation

This is a means by which the measure of accuracy of a forecasting method is determined. It is used by getting the average of the forecasting methods as follows;

**(MAD)** = sum of Forecasting Errors/ Number of Forecasts

## Mean Square Error

This is used in the checking for accuracy in forecasting in regards to large forecasting errors. This is due to the fact that large errors are taken more seriously than smaller ones. This method can also be referred to as the mean square error because it uses the average of the square of its forecasting errors as follows;

** (MSE)** = Sum of Square of Forecasting Errors/ Number of forecasts

## Queues and their significance

Queues can be termed as waiting lines and are used daily in day to day life. It can also be termed as the study of queuing models to represent various aspects of the queuing systems that arise in practice (Hiller & Hiller, 2010). Queues have serious implications especially if customers have to wait for long in them and thus the need for queue models used to determine how much service capacity should be provided to a queue to avoid excessive waiting.

## Queuing models

In the queuing model, there is the basic queuing system whereby customers arrive individually and receive some kind of service and if he can not be served directly, he joins a queue to await for service. One or more people provide the service for the customers in this model.

The times between arrivals to a queuing system are known as the interval times. And this are calculated through: the estimation of the expected number of arrivals per unit time which is the mean arrival rate and the estimation of the form of the probability distribution of inter arrival times.

A queue can be defined as the place which customers wait until being served and the number of customers in the queue can be termed as its size while the number of customers in the system is referred to as the number of customers in the queue plus the number currently being served.

It is also good to note that the order in which members of the queue are selected to begin service is referred to as its discipline. There is also Infinite queues which are referred to as queues with an unlimited number of customers who can be held in the queue and finite queue which is the opposite of infinite queues. Finite queues are queues that their customer carrying capacity is limited to a specific number.

Most organizations use queuing systems for efficient service delivery and for orderliness. There are various examples of queuing systems. The following are some of those examples;

## Examples of queuing systems

### Commercial service system

In the commercial type of service system, an organization takes the mandate of providing customers who are not from within the said organization with various services. These customers come, are served and return to their various destinations. Examples include institutions such as the Barber shop where the customers are people and the server is the barber. Another example can be the plumbing services where the clogged pipes can be termed as the customers and the server is the plumber (Hiller & Hiller, 2010).

### Internal service system

This is a queuing system which customers receiving service are internal to the organization providing the service. Examples include the Mainframe computer where the customers are employees and the server is the computer and plumbing Inspection station where the customers are the items and the server is the inspector.

### Transportation Service Stations

The transportation service system is a system which referrers to the involvement of transportation services. This is such that either the customer or the server is a vehicle. Examples include the highway tollbooth where the cars can be termed as the customers while the server is the cashier. Another example is the airline service where the customers are the people using the airline while the server is the airplane (Hiller & Hiller, 2010).

## Conclusion

In conclusion, as Hillier et al. (2010) states, management decisions can determine the success or failure of a firm. Thus, it is mandatory for the necessary for proper analysis and forecasting to be performed before decisions can be reached at. It is also important to note here that not all statistical tools for analysis into the decision making process have been discussed by me in this paper.

This is a wide subject area that could not be covered in one assignment. However, it is an essential matter to have customers in mind whenever models are developed in any level because they are the end consumers of the company’s product and services.

## References

Gale, D (1960). *The Theory of Linear Economic Models*, New York, NY: McGraw-Hill.

Hillier, M., & Hillier, S.(2010). *Introduction to Management Science: A Modeling and Case Studies Approach with Spreadsheets*, (4th Ed), New York, NY: McGraw-Hill/Irwin Publishing Company

Karlin, S. (1959). ‘Mathematical Methods and Theory in Games**,’ ***Programming and Economics*, vol. 1, Addison-Wesley

Strayer, K. (1989). *Linear Programming and Applications,*New York, NY: Springer