7.7. Extension of the Example: Adding Almond Rasmalai

So, just to get a little bit more practice at this, let us extend the example. So, now, in addition to our barfi and halwa which have the same details that before a 100 rupee profit for barfis, 600 for halwa we added almond rasmalai which gives us a much higher profit of 1300. An almond rasmalai actually people are willing to buy in an unlimited quantity. So, the earlier demand for barfis and halwa is the same, 200 and 300, but almond rasmalai can be sold in unlimited amounts.

This chunk introduces a new product, almond rasmalai, into the example of a sweet shop's production strategy. The key points are that almond rasmalai has a significantly higher profit margin of 1300 rupees per box and can be sold in unlimited quantities. This contrasts with the existing products, barfis and halwa, which have fixed maximum demand (200 and 300 boxes respectively). Hence, the introduction of almond rasmalai changes the dynamics of the optimization problem, making it more complex.

Imagine running a bakery that sells cupcakes and cookies, both of which have a maximum order (like barfis and halwa). Then, you start offering a new type of cake that's highly popular, with no limits on how much can be sold. This is similar to how almond rasmalai can be produced without worrying about running out of customers.

Unfortunately, we have not got any new staff, so we are still restricted to producing a total of 400 boxes and now we have an extra constraint which is in terms of what we can do with the milk that we get. So, with the milk that we get we can either makes 600 boxes of halwa in a day or to want a rasmalai or any combination of this. So, effectively a box of rasmalai requires three times as much as milk as a box of halwa.

In this chunk, we consider the operational constraints regarding the production of sweets. Even though almond rasmalai can be sold in unlimited quantities, the production is still capped at a total of 400 boxes due to staff limitations and milk availability. The milk constraint is significant: one box of rasmalai uses three times the milk of one box of halwa. This means that if you choose to produce more rasmalai, you will produce less halwa because both use the same milk supply.

Think of a scenario where a restaurant can only cook a certain number of dishes due to limited kitchen space or staff. If one dish (like a complicated main course) takes much longer or requires more resources (like cook time or ingredients), the restaurant must choose wisely how many to prepare alongside simpler dishes.

So, if we look at the equation now, the objective function now has an extra quantity which is the profit we get from rasmalai. So, we have an extra variable, we have b and h which are as before barfi and halwa and r for rasmalai, so we can get 100 rupees per box of barfi, 600 rupees per box of halwa and 1300 per box of rasmalai.

This chunk explains how the objective function of the linear programming problem now incorporates a new variable for rasmalai (r). Therefore, the overall profit equation now includes profits from barfis and halwas along with rasmalai, represented as 100b + 600h + 1300r. This modification reflects the expanded goal of maximizing profits considering the new product.

Returning to the bakery example, think of how you adjust your sales projection. If you add a new item with higher profit margins, you have to update your calculations and projections to include this new revenue stream in your financial forecast.

The constraints on barfi and halwa demand of the same and there is no constraint for rasmalai, so we do not add anything there. The total production now includes the rasmalai, so all three together must be blue 400 and this expressives the milk constraint.

In this chunk, the constraints are detailed. While barfi and halwa maintain their previous production limits, rasmalai does not have a demand limit. However, the total production still cannot exceed 400 boxes, and the amount of halwa produced can't exceed limits set by the amount of available milk, represented as h + 3r ≤ 600. This shows how to formulate a complex linear programming problem, incorporating multiple constraints.

Imagine planning a dinner menu. You may have fixed numbers of certain dishes (like fish and chicken) while introducing a third dish (like a salad) that you can make as much as you want; however, you also have to factor in limited kitchen resources (like oven space or preparation time) that limit how much of everything you can cook.

So, in this case if we draw this picture, now it becomes a three dimensional geometrical object, because we have three axis b, h and now r coming out in the vertical direction and the constraints now instead of the lines, they become these plans. So, the kind of become now instead of a polygon, we have what is called polytope, we have a three dimensional object.

This chunk discusses how the introduction of a third product makes the model more complex, shifting from two dimensions to three dimensions. Now, we visualize the solution space as a polytope instead of a polygon. Constraints are no longer simply lines but planes in space functioning to limit the allowable combinations of barfi, halwa, and rasmalai.

Consider a scenario of baking with three different cake flavors. The graph in two dimensions could show combinations of two flavors, but now it expands into three dimensions when you add a third flavor. Visualizing becomes much more complex as you try to find the right combination that meets all tastes while remaining feasible, just like our sweets production chart.

And again we will find that you get the optimum at some corner and the optimum happens to be in this case that you make zero barfis, you make a full amount 300 halwa and then you make the rest for rasmalai and then you get a profit of 3 lakhs 10,000.

This final chunk summarizes the conclusions of the extended example, identifying that the optimal production strategy is to produce no barfis, 300 halwas, and the remainder as almond rasmalai. This specific combination yields the highest profit, illustrating how to arrive at an optimal solution through exploration of the feasible region defined by the constraints.

Think of a game strategy where you have several tactics available. You might analyze gameplay data and find that the best combination of moves (like certain attacks or defenses) leads to the maximum score; similarly, in our sweet shop, the right mix of products leads to the highest profit.

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