7.8. Three-Dimensional Geometrical Representation

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.

In this chunk, we introduce that when we have three variables, we can visualize the problem in a three-dimensional space. Each variable will represent an axis in this space. For this example, b (barfis), h (halwa), and r (rasmalai) are our three variables. Unlike in two dimensions where relationships are represented by lines, here in three dimensions, relationships are represented by planes. The area of possible solutions is now a polytope, a geometric shape confined by multiple planes rather than just a polygon.

Think of trying to find the best combination of ingredients for a smoothie. If each ingredient (like fruit, yogurt, and protein powder) corresponds to an axis in 3D space, then every mixture represents a point in that space. Just as you can't make a smoothie with negative amounts of ingredients, you can only choose within the constraints we set for barfis, halwa, and rasmalai.

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.

In this chunk, we emphasize the importance of locating optimal solutions in the feasible space. The solution to our production problem involves analyzing the vertices (corners) of our polytope. Here, the optimal production schedule is reached when no barfis are made, 300 halwas are produced, and the remainder is allocated to rasmalai. This provides the highest possible profit. Thus, in linear programming, it is common for the best solution to lie at the vertex of the feasible region.

Imagine a treasure hunt where the treasure can only be at specific checkpoints. If you follow the rules and look at the corners of a map (vertices), you might find the treasure hidden at one of these points. Similarly, in linear programming, the maximum profit is often found at specific boundary points rather than within the middle of the feasible region.

So, is this really our best thing? So, remember that if I do this, I get 300 into 600, so I get 1 lakh 80,000 from here and you get 1 lakh 30,000 for here and thus how we get a 3 lakhs 10,000. So, is it how do I know that this answer is actually the best?

Here, we question the solution to ensure it is indeed the optimal outcome. By calculating the profit generated from the chosen production schedule—300 halwas yielding 1 lakh 80,000 and the profit from rasmalai adding up to a total of 3 lakhs 10,000—we solidify the argument that this is the best solution as it adheres to our constraints while maximizing profit. This process of verification provides certainty that we have explored feasible solutions and identified the maximum.

Think of a manager evaluating sales performance. They want to confirm that a specific sales strategy yielded the highest profit by comparing all possible combinations of products sold. Just as the manager checks sales data and strategies to find out the most profitable approach, we review our calculated profits to ensure our production plan is optimal.

It turns out that fortunately this is not a coincidence, it turns out you can always construct such a combination. So, if I have my constraints C 1, C 2, so these are my constraint equations, I can always find some combination.

In this part, we discover that finding an optimal solution is not just luck; it's backed by mathematical principles. The optimal production schedule can be determined by establishing combinations of constraints—each constraint contributes to defining the limits of our feasible region. By manipulating these constraints, we can gauge the maximum profit possible, affirming it as the optimum through mathematical validation.

Consider a chef tasked with creating a new dish using limited ingredients. By knowing the recipe's constraints about quantities and how they combine, the chef can experiment by adjusting amounts until they discover the perfect balance that maximizes flavor. Similarly, in linear programming, adjusting constraints allows us to pinpoint the maximum outcome.

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