Understanding the Assignment Model and Queuing theory: A Comprehensive Guide for Beginners

Assignment Model

Assignment model is like transportation model except you decide whether or to assign a source to estimation (or employee to a task). Decision variable is binary suppose you have three employees and three tasks. How many different possible assignments are there? Assignment problem is one of the special cases of the transportation problem. It involves assignment of people to projects, jobs to machines, workers to jobs and teachers to classes etc., while minimizing the total assignment costs. One of the important characteristics of assignment problem is that only one job (or worker) is assigned to one machine (or project). Hence the number of sources is equal the number of destinations and each requirement and capacity value is exactly one unit. Although assignment problem can be solved using the techniques of Linear Programming or the transportation method, the assignment method is much faster and efficient.

This method was developed by D.Konig, a Hungarian mathematician and is therefore known as the Hungarian method of assignment problem. In order to use this method, one needs to know only the cost of making all the possible assignments. Each assignment problem has a matrix (table) associated with it. Normally, the objects (or people) one wishes to assign are expressed in rows, whereas the columns represent the tasks (or things) assigned to them. The number in the table would then be the costs associated with each particular assignment.

It may be noted that the assignment problem is a variation of transportation problem with two characteristics.

1. The cost matrix is a square matrix
2. The optimum solution for the problem would be such that there would be only one assignment in a row or column of the cost matrix.

Constraints 

1. Each assignment must get at most I assignee.
2. Each assignee must get at most I assignment.
3. Non negativity constraint
4. Integer constraint (use integer programming)

Application Areas of Assignment Problem 

1. In assigning machines to factory orders.
2. In assigning sales/marketing people to sales territories.
3. In assigning contracts to bidders by systematic bid evaluation.
4. In assigning teachers to classes.
5. In assigning accountants to accounts of the clients

The assignment problem can be solved by the following four methods

1. Enumeration method - In this method, a list of all possible assignments among the given resources and activities is prepared. Then an assignment involving the minimum cost, time or distance or maximum profits is selected. If two or more assignments have the same minimum cost, time or distance, the problem has multiple optimal solutions. This method can be used only if the number of assignments is less. It becomes unsuitable for manual calculations if number of assignments is large.
2. Simplex method - The graphical method is capable of solving problems having a maximum of two variables. Hence, this method is used which can solve LP problems with any no. of variable or constraints it is geared towards solving optimization problems which have constraints of less than or equal to type.
3. Transportation method 
4. Hungarian method - There are various ways to solve assignment problems. Certainly, it can be formulated as a linear program (as we saw above), and the simplex method can be used to solve it. In addition, since it can be formulated as a network problem, the network simplex method may solve it quickly. However, sometimes the simplex method is inefficient for assignment problems (particularly problems with a high degree of degeneracy).

Queuing theory

Queuing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide service. Queuing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the (back of the) queue, waiting in the queue (essentially a storage process), and being served at the front of the queue. 

The theory permits the derivation and calculation of several performance measures including the average waiting time in the queue or the system, the expected number waiting or receiving service, and the probability of encountering the system in certain states, such as empty, full, having an available server or having to wait a certain time to be served. Queuing theory has applications in diverse fields, including telecommunications, traffic engineering, computing and the design of factories, shops, offices and hospitals. Applications are frequently encountered in customer service situations as well as transport and telecommunication. Queueing theory is directly applicable to intelligent transportation systems, call centers, PABXs, networks, telecommunications, and server queueing, mainframe computer of telecommunications terminals, advanced telecommunications systems, and traffic flow. For example, "G/D/1" would indicate a General (may be anything) arrival process, a Deterministic (constant time) service process and a single server. More details on this notation are given in the article about Queueing models.

Arrivals → Queue → Service →Departures

Here are details of four queuing disciplines

1. First in first out - This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
2. Last in first out - This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first it is also known as a stack.
3. Processor sharing - Customers are served equally. Network capacity is shared between customers and they all effectively experience the same delay.
4. Priority - Customers with high priority are served first.

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Sandeep Ghatuary

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