7.5. Simplex Algorithm Overview

So, we can look at a node general formulation of such constraint optimization problems in the framework of what is called linear programming. So, in linear programming we are given some variables, some quantities that we want to calculate and then there are some linear functions that constraint these quantities.

Linear programming is a method used to optimize a certain outcome, given some constraints. In such problems, we deal with variables (quantities we can control) subject to certain limits (constraints). The mathematical representation involves linear functions, which means we only deal with first-degree terms without any exponent higher than one.

Imagine planning a trip: you have a budget (constraint) and want to maximize the number of sights you can see (objective). You can only choose from places that fit within your budget, hence illustrating the essence of linear programming.

So, suppose we are running a sweets shop called grandiose sweets and we tell sell two types of sweets, barfis and halwa. Now, we know that each box of barfis earns a profit of 100 rupees and each box of halwa earns a profit of 600 rupees.

In this example, we have a sweets shop selling two products, barfis and halwa. The profits from each product are provided, which allows us to establish a clear objective function - maximizing profit. It's essential to identify what variables (number of barfis, number of halwa) influence our outcome and then apply the corresponding profit values to these variables.

Think of a lemonade stand: if you know that each cup of lemonade sells for a dollar and each cookie sells for two dollars, your goal could be to determine how many of each to make to earn as much money as possible.

Now, we also know that on a given day we cannot sell more than 200 barfi boxes, we cannot sell more than 300 halwa boxes and together the staff can only produce 400 boxes.

In addition to our goal of maximizing profit, we have several constraints that limit our production capabilities. These constraints must be represented as mathematical inequalities that dictate the permissible quantities of barfis and halwas we can produce based on sales limits and staff capacity.

Imagine you're planning a party with a specific budget. You can't buy more than a certain number of pizzas or drinks due to both budget constraints and storage space limitations. These limits dictate how much of each item you can actually afford.

So, our objective is to maximize the profit, so this is the profit, so this is what we are going to maximize, the quantity 100 b plus 600 h.

Here, we are constructing the objective function, which in our case is the total profit made from selling barfis and halwas. This function combines both variables (the number of barfis and halwas) and their respective profit contributions into a single expression that we want to maximize.

Returning to the lemonade stand, if you sell 10 cups and 5 cookies, your total income could be calculated as $10 +$10 = $20. The 'profit function' interprets the income from both items to determine the best-selling strategy.

So, overall this gives us what we can call a feasible region, any point in this region represents a combination of b value and h value which means all the constraints.

The feasible region is the area within which all the constraints of the problem are satisfied. Any combination of barfis and halwas that falls within this area will yield valid solutions. Graphically representing this region helps us visualize the limits of our production capabilities.

If you plot the constraints of your party budget on a graph, the area that shows how many pizzas and drinks you could afford within your budget would represent the feasible region for your party supplies.

It turns out that this optimum value happens at some corner of this feasible region and this is generally the case, one can argue that the optimal value will always occur at a vertex.

In your feasible region, the maximum or minimum values of the objective function are typically found at the vertices (corners) of the region. This is a critical observation in linear programming that aids in simplifying the search for the optimal solution.

If you’re designing a triangular playground within a set perimeter, the best way to maximize space utilization would be to design it around the corners of the area provided, ensuring you’re making the most of the available space.

This is how the very famous simplest simplex algorithm works. So, what it does is, it constructs this feasible region which is bounded by constraints and then it picks a vertex.

The simplex algorithm automates the process of finding the optimal solution by iterating through the vertices of the feasible region, evaluating the objective function at each vertex until it identifies the highest (or lowest) value. This method, while not guaranteed to be the fastest in theory, is practical and effective in solving linear programs.

Think of a treasure hunt: you navigate to each point (vertices) based on clues (constraints) until you discover which point has the treasure (optimal solution), ensuring you cover all possibilities without missing out.

Now, there are theoretically efficient algorithms, such as the interior point methods, but simplex is probably easier to program.

While other algorithms exist which may have theoretical advantages, the simplicity and practicality of the simplex algorithm make it a go-to solution for many linear programming problems. It strikes a balance between efficiency and ease of implementation in real-world applications.

It's akin to using a straightforward map to reach a destination instead of opting for a complex GPS system, which, although more advanced, might take longer to set up.

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