7.2. Formulating the Linear Program

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. So, linear function remember of a variable x is something of the form a x plus b. So, it has no x square x cube term, it is all linear, so we have a x plus b.

Linear programming is a method used to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. It involves defining variables that represent quantities we want to optimize and establishing linear function constraints on these variables. A linear function is defined as any function that can be expressed in the form 'a * x + b', where 'x' is a variable and 'a' and 'b' are constants. The key is that functions in linear programming must not include any terms that are not linear, meaning you cannot have terms like x² or x³.

Think of it like budgeting your monthly expenses: you can only spend a certain amount based on your income (the constraints). For instance, if your income is fixed and you wish to spend on food, rent, and entertainment (variables), you need to calculate how to allocate your income across these expenses without exceeding it.

So, in general then if we have multiple variables x y z, then we could have a constraint of the form a x plus b y plus something is less than equal to some constant or a x plus b y plus something is greater than equal to some constant. And then these are the constraints on the values that the variables can take and now our aim is to optimize some quantity, we want to minimize or maximize some cause or some weights.

In linear programming, once the variables (like x, y, and z) are defined, we establish constraints which are inequalities representing the limits on these variables. For example, we might say that 'a * x + b * y must be less than or equal to a specific value'. The goal is to optimize (maximize or minimize) an objective function, which is also a linear equation typically expressed in the form of these variables. Identifying the right constraints is crucial as they define the feasible region within which the solution must lie.

Imagine you're planning a trip. You have multiple destinations (variables) to visit, but you have limited time (constraints). If visiting more places means spending less time at each, the constraints ensure you don't exceed the time available for the trip. Maximizing your destinations while respecting the time limits reflects how we use constraints in linear programming.

So, the best way to understand linear programming to begin with is to look at an example. 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. So obviously, make sense to make more halwa than barfis.

To illustrate linear programming practically, consider a sweet shop producing two types of sweets: barfis and halwas. Each box of barfis brings a profit of ₹100, while each box of halwa brings ₹600. The problem becomes one of figuring out how many boxes of each to produce in order to maximize profits while adhering to certain production and sales constraints. The sales limits and production limitations frame the linear programming model.

Think of this like running a bakery where you can make different types of cakes. If you know that chocolate cakes sell for more, you might prioritize making them over vanilla cakes. However, you still have to consider how many of each you can bake given your oven capacity and ingredients available. This decision-making framework aligns closely with linear programming.

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 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. So, b and h must lie between 0 and 200 and 0 and 300 respectively and together they can be no more than 400.

In this scenario, the objective function—the mathematical expression we want to maximize—is defined as '100b + 600h', where 'b' is the number of boxes of barfis and 'h' is the number of boxes of halwas. Given the constraints (b ≤ 200, h ≤ 300, and their total production ≤ 400), we create a set of inequalities that define the limits within which 'b' and 'h' must operate to abide by both production capacity and market demand.

It's similar to planning a party. If you have a limited budget (the constraints) and want to serve appetizers and drinks (the variables), you must figure out how many of each to buy so you stay within budget while providing enough of both to keep guests happy. The balance between these items to maximize enjoyment would be like maximizing the profit in our mathematical model.

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