## Dealing With Exceptional Cases Of Transportation And Assignment Problems For Quantitative Techniques For Business Decisions Mcom Sem 2 Delhi University Notes

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### Dealing With Exceptional Cases Of Transportation And Assignment Problems For Quantitative Techniques For Business Decisions Mcom Sem 2 Delhi University Notes

Dealing With Exceptional Cases Of Transportation And Assignment Problems For Quantitative Techniques For Business Decisions MCOM Sem 2 Delhi University : Transportation and assignment models are special purpose algorithms of the linear programming. The simplex method of Linear Programming Problems(LPP) proves to be inefficient is certain situations like determining optimum assignment of jobs to persons, supply of materials from several supply points to several destinations and the like. More effective solution models have been evolved and these are called assignment and transportation models.

The transportation model is concerned with selecting the routes between supply and demand points in order to minimize costs of transportation subject to constraints of supply at any supply point and demand at any demand point. Assume a company has 4 manufacturing plants with different capacity levels, and 5 regional distribution centres. 4 x 5 = 20 routes are possible. Given the transportation costs per load of each of 20 routes between the manufacturing (supply) plants and the regional distribution (demand) centres, and supply and demand constraints, how many loads can be transported through different routes so as to minimize transportation costs? The answer to this question is obtained easily through the transportation algorithm.

Similarly, how are we to assign different jobs to different persons/machines, given cost of job completion for each pair of job machine/person? The objective is minimizing total cost. This is best solved through assignment algorithm.

**Uses of Transportation and Assignment Models in Decision Making**

The broad purposes of Transportation and Assignment models in LPP are just mentioned above. Now we have just enumerated the different situations where we can make use of these models.

Transportation model is used in the following:

- To decide the transportation of new materials from various centres to different manufacturing plants. In the case of multi-plant company this is highly useful.
- To decide the transportation of finished goods from different manufacturing plants to the different distribution centres. For a multi-plant-multi-market company this is useful.
- To decide the transportation of finished goods from different manufacturing plants to the different distribution centres. For a multi-plant-multi-market company this is useful. These two are the uses of transportation model. The objective is minimizing transportation cost.

Assignment model is used in the following:

- To decide the assignment of jobs to persons/machines, the assignment model is used.
- To decide the route a traveling executive has to adopt (dealing with the order inn which he/she has to visit different places).
- To decide the order in which different activities performed on one and the same facility be taken up.

In the case of transportation model, the supply quantity may be less or more than the demand. Similarly the assignment model, the number of jobs may be equal to, less or more than the number of machines/persons available. In all these cases the simplex method of LPP can be adopted, but transportation and assignment models are more effective, less time consuming and easier than the LPP.

### Dealing With Exceptional Cases Of Transportation And Assignment Problems For Quantitative Techniques For Business Decisions MCOM Sem 2 Delhi University Notes

Dealing With Exceptional Cases Of Transportation And Assignment Problems For Quantitative Techniques For Business Decisions MCOM Sem 2 Delhi University : The **assignment problem** is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics. It consists of finding a maximum weight matching (or minimum weight perfect matching) in a weighted bipartite graph.

In its most general form, the problem is as follows:

- The problem instance has a number of
*agents*and a number of*tasks*. Any agent can be assigned to perform any task, incurring some*cost*that may vary depending on the agent-task assignment. It is required to perform all tasks by assigning exactly one agent to each task and exactly one task to each agent in such a way that the*total cost*of the assignment is minimized.

If the numbers of agents and tasks are equal and the total cost of the assignment for all tasks is equal to the sum of the costs for each agent (or the sum of the costs for each task, which is the same thing in this case), then the problem is called the *linear assignment problem*. Commonly, when speaking of the *assignment problem* without any additional qualification, then the *linear assignment problem* is meant.

#### The formal definition of the **assignment problem** (or **linear assignment problem**) is

- Given two sets,
*A*and*T*, of equal size, together with a weight function*C*:*A*×*T*→**R**. Find a bijection*f*:*A*→*T*such that the cost function:

is minimized.

Usually the weight function is viewed as a square real-valued matrix *C*, so that the cost function is written down as:

The problem is “linear” because the cost function to be optimized as well as all the constraints contain only linear terms.

The problem can be expressed as a standard linear program with the objective function

subject to the constraints

The variable represents the assignment of agent to task , taking value 1 if the assignment is done and 0 otherwise. This formulation allows also fractional variable values, but there is always an optimal solution where the variables take integer values. This is because the constraint matrix is totally unimodular. The first constraint requires that every agent is assigned to exactly one task, and the second constraint requires that every task is assigned exactly one agent.

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### Dealing With Exceptional Cases Of Transportation And Assignment Problems For Quantitative Techniques For Business Decisions Mcom Sem 2 Delhi University Notes

Dealing With Exceptional Cases Of Transportation And Assignment Problems For Quantitative Techniques For Business Decisions MCOM Sem 2 Delhi University : Transportation problem is a particular class of linear programming, which is associated with day-to-day activities in our real life and mainly deals with logistics. It helps in solving problems on distribution and transportation of resources from one place to another. The goods are transported from a set of sources (e.g., factory) to a set of destinations (e.g., warehouse) to meet the specific requirements. In other words, transportation problems deal with the transportation of a single product manufactured at different plants *(supply origins)* to a number of different warehouses *(demand destinations). *The objective is to satisfy the demand at destinations from the supply constraints at the minimum transportation cost possible. To achieve this objective, we must know the quantity of available supplies and the quantities demanded. In addition, we must also know the location, to find the cost of transporting one unit of commodity from the place of origin to the destination. The model is useful for making strategic decisions involved in selecting optimum transportation routes so as to allocate the production of various plants to several warehouses or distribution centers.

Suppose there are more than one centers, called **‘origins’** , from where the goods need to be transported to more than one places called **‘destinations’** and the costs of transporting or shipping from each of the origin to each of the destination being different and known. The problem is to transport the goods from various origins to different destinations in such a manner that the cost of shipping or transportation is minimum.

Thus, the transportation problem is to transport various amounts of a single homogenous commodity, which are initially stored at various origins, to different destinations in such a way that the transportation cost is minimum.

The objective of the transportation model is to determine the amount to be shipped from each source to each destination so as to maintain the supply and demand requirements at the lowest transportation cost.

For example: A tyre manufacturing concern has *m* factories located in *m* different cities. The total supply potential of manufactured product is absorbed by *n* retail dealers in *n* different cities of a country. Then, transportation problem is to determine the transportation schedule that minimizes the total cost of transporting tyres from various factory locations to various retail dealers.

The transportation model can also be used in making location decisions. The model helps in locating a new facility, a manufacturing plant or an office when two or more number of locations is under consideration. The total transportation cost, distribution cost or shipping cost and production costs are to be minimized by applying the model.

There is a type of linear programming problem that may be solved using a simplified version of the simplex technique called **transportation method**. Because of its major application in solving problems involving several product sources and several destinations of products, this type of problem is frequently called the **transportation problem**. It gets its name from its application to problems involving transporting products from several sources to several destinations. Although the formation can be used to represent more general assignment and scheduling problems as well as transportation and distribution problems. The two common objectives of such problems are either (1) minimize the cost of shipping *m* units to *n* destinations or (2) maximize the profit of shipping *m* units to *n* destinations.

Let us assume there are *m* sources supplying *n* destinations. Source capacities, destinations requirements and costs of material shipping from each source to each destination are given constantly. The transportation problem can be described using following linear programming mathematical model and usually it appears in a transportation tableau.

There are three general steps in solving transportation problems.

We will now discuss each one in the context of a simple example. Suppose one company has four factories supplying four warehouses and its management wants to determine the minimum-cost shipping schedule for its weekly output of chests. Factory supply, warehouse demands, and shipping costs per one chest (unit) are shown in Table 7.1

** Table 7.1 ”Data for Transportation Problem”**

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### Dealing With Exceptional Cases Of Transportation And Assignment Problems For Quantitative Techniques For Business Decisions MCOM Sem 2 Delhi University Notes

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