7.9. Justifying Optimum Profit

The objective is to maximize the profit, so this is the profit, so this is what we are going to maximize, the quantity 100 b + 600 h and we cannot choose obviously, if b and h can vary, then we can make them as large as we want, but there are these constraints.

In this scenario, we aim to maximize profit by balancing the production of barfis (b) and halwa (h). The profit function is represented as 100x + 600y, indicating that 100 rupees are earned for each box of barfis and 600 rupees for each box of halwa. However, our choices are constrained by the maximum sales limits and the total production capacity of 400 boxes per day. This means that while we want to make the numbers as high as possible for profit, we must adhere to these limitations.

Think of a bakery that wants to maximize its income from cakes and pastries. If the bakery can only make a certain number of items (like a maximum of 200 cakes and 300 pastries) and has to work within that limit, the bakery must find the best combination that generates the most sales.

So we can visualize this quite easily in this case, because there are two quantities basically b and h that we want to compute and we are given a set of facts about them. So we know that b lies between 0 and 200, so the range for b is anything in this region.

In linear programming, we can visualize the constraints by plotting them on a graph, creating a feasible region where all possible combinations of b and h fulfill the constraints defined earlier. For instance, if b can range from 0 to 200 and h from 0 to 300, we can draw this as a box on the graph, helping us identify the area where feasible combinations exist.

Imagine trying to find a spot to park your car—you're limited to spaces marked by lines on the ground. Wherever these lines meet and form a box is like the feasible region: you can only park your car within these bounds.

When that line actually leaves the region, at that corner vertex whether at pcm turns bends down, we find that we can get 1 lakh 90,000 and it turns out that this is actually going to be the optimum value.

The optimal solution in linear programming typically occurs at the vertices of the feasible region. As we adjust profit lines (for example, representing different profit levels), we find that the maximum profit, in this case 190,000 rupees, occurs at one of these corners. This principle helps simplify the search for the optimal solution since we only need to evaluate these vertices instead of every single possible combination within the feasible region.

Think of climbing a mountain where all you want to do is reach the peak. You might have multiple paths (combinations of production), but the peak (the optimal profit) is always at a specific point. Just like the peak represents the maximum height you can achieve, the vertices of the feasible region represent maximum profit you can aim for.

So, the profit is exactly 100 b + 600 h, 1300 r, we are trying to maximize this and what our constraints tell us is that in the feasible region, this value can be no more than three ((Refer Time: 16:05)) should be, 3 lakhs 10,000.

Finally, to verify the optimal profit found, we look at the constraints. By manipulating the constraints in a way that reflects the profit function, we deduce that the maximum possible profit achievable under these constraints doesn't exceed 310,000. This confirmation process validates the solution's correctness, ensuring it doesn't just luck into a high number but actually aligns with mathematical principles governing our defined limits.

Consider a sports player who wants to score as many points as possible. They must follow the rules of the game (constraints) to score maximum points. By analyzing their play and the rules, they confirm that the points they achieve are indeed the best they can do within the game rules.

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