When we are packing something in a box or a suitcase or a cupboard, first we have to decide how we are going to pack. How many items can be fitted into that fixed space? A good example of this is, before you go to school, you try to pack everything you need (like your books, notebooks, geometry box, sports equipment, food and water bottle) into your school bag. At that time, you are very clear that your books should not be damaged and you are able to carry everything yourself. Think! The same rules apply to posting a package to a friend or family member or others.
Apart from these, the packaging methods are used in many cases such as cutting of sheets, glass, paper, wood, cloth or other materials and room allotment, seating arrangement in the particular space provided, parking vehicles with proper lanes and saving data in hard disk, CD, pen drive and so on.
Using some of the packing methods from the following situations and examples, let us try to understand how best to fit the items into the space in the containers or in rooms or in boxes etc.,
1. Packing Approaches - Fractional method
Here, when we fill items in bags or in containers, we determine the weight, value and number of each item with the condition that the total weight of the container is less than or equal to a given limit and the total value is as large as possible. Fractional method uses the technique of buying things fractionally and admits buying of more items within a given budget. Let us learn more about this approach from the following situation.
Suppose that you want to buy some vegetables and fruits that are given in the Fig. 7.20 with their weights and price and that you have a bag that capacity of carrying 15 kg. The objective is to buy more items within your budget of ₹550 and also that weight should not exceed 15 kg.
You cannot buy all the items because if you calculate the total weight of all the items, then it would be greater than 15 kg (maximum capacity of your bag is 15 kg). So, let us try some approaches, to find how you can select more items so that you will buy them with maximum price within your budget of ₹550. For that let us tabulate the items with their weights and price you want to buy.
Approach I - Selecting items with Maximum Price
In this approach, we select the items according to the maximum price. Here the maximum price in the table is ₹150/-. Now, let us tabulate to find the total price and how much can we buy vegetables and fruits within our budget and not exceeding 15kg.
Here, 3 kg of papaya is enough as the total weight reaches 15 kg. So, 3 kg of papaya costs ₹52.50. Hence, in this approach we will spend maximum ₹472.50 to buy 15 kg of vegetables and fruits.
Approach II - Selecting items with Minimum Weight
In this approach, we select the items according to the minimum weight. Here, we can select more and more items. Now, let us tabulate to find the total price and how much can we buy vegetables and fruits within our budget and not exceeding 15kg.
Here, 1 kg of Sapotta is enough to complete 15 kg with minimum price of ₹30 per kg.
Hence in this approach, we will spend maximum ₹475 to buy 15 kg of vegetables and fruits.
Approach III - Finding the Maximum price to Weight Ratio.
In this approach, we select the items according to the maximum price to weight ratio (find the rate of 1kg). Now, let us tabulate to find the total price and how much we can buy vegetables and fruits within our budget and not exceeding 15 kg.
In this approach, we can buy all vegetables and fruits except papaya as we need with maximum price within our budget and not exceeding 15 kg. Comparatively, in the II approach we can buy more items but spend minimum amount only. So, we can say third approach is best one. Isn’t it?