How to solve Complex Arrangement Problems using Tables and Matrix

Complex arrangement problems are very much similar to linear arrangement problem except for some extra dimensions in them.

Following example will:

There are five villages, A, B, C, D and E. Two of these have a post office, three have schools and three are accessible by road. Two have a population of more than 5000 each, two have population between 2000 and 5000 each, and one has a population of less than 2000. Two of these villages have electricity in addition to certain other facilities such as schools and accessibility by road. One village with population of less than 2000 also has a school but neither has a post office nor is accessible by road while each village with population of more than 5000 has a school. Of the two the village having population between 2000 and 5000, only one has a post office and is accessible by road. A is accessible by road. B has a population of more than 5000, D has a school and is accessible by road but does not have a post office while E has a school but is not accessible by road.

As you can notice through the above illustration that rather than one characteristic feature it has 3, unlike linear arrangement problems which have straightforward information. These extra features make these problems complex and not the solution. The solution to these problems is quite simple and there are two methods to acquire it. These are

• Matrix Method
• Table Method

Now the next step is to start reading putting up the direct information associated with the object given in the problem by using ‘+’ for yes and * for no. Let’s start doing it for the above given information.It’s been given directly that

• A is accessible by road.
• B has a population of more than 5000.
• D has a school and is accessible by road but does not have a post office.
• E has a school but is not accessible by road.

Now using the above bold points we need to estimate the link with what other information is given in the problem that relates them with other characteristics.

It’s been given that the village with population more than 5000 has a school which implies B has a school and this point also has another implication that there are total three schools and we just established on in B and other two are in D and E which means either D or E has a population greater than 5000. To figure out which one has the population we can use another fact provided above i.e. one village with the population of less than 2000 also has a school but neither has a post office nor is accessible by road. Now, since D is accessible by road, therefore, it can’t be the one with the population less than 2000 and hence, E is the one with population less than 2000. Also, A and C does not have schools. Now let’s add in this information in our matrix.

As you can see we have almost covered more than half of the matrix and we can clearly find that both A and C are the villages have the population between 2000 & 5000. And only one of them has a post office and accessibility by road. Since A is accessible by road, therefore, it must have a post office. Also, three villages have road accessibility we have established two of them and one is left which should be B as only one the village having population between 2000 and 5000, only one has a post office and is accessible by road thus C can’t be the one and thus also implies that B has the post office as well. Also, two of these villages have electricity in addition to certain other facilities such as schools and accessibility by road and the only suited options that fulfill these criteria is B and D. Hence, the required matrix is: