Maximizing gross profit A stationery company makes two types...

Maximizing gross profit A stationery company makes two types of notebooks: a deluxe notebook with subject dividers, which sells for $$ 4.00,$ and a regular notebook, which sells for 3.00 dollars . The production cost is 3.20 dollars for each deluxe notebook and $$ 2.60$ for each regular notebook. The company has the facilities to manufacture between 2000 and 3000 deluxe and between 3000 and 6000 regular notebooks, but not more than 7000 altogether. How many notebooks of each type should be manufactured to maximize the difference between the selling prices and the production costs?

Key Concepts

Profit Contribution Analysis

Profit contribution analysis involves calculating the net profit generated by each unit produced, which is determined by subtracting the production cost from the selling price. This concept helps in understanding how each additional unit contributes towards the overall profit, guiding decision-makers in prioritizing production towards items with higher net benefits.

Feasible Region

The feasible region is the set of all points that satisfy the model's constraints, representing all possible solutions to the linear programming problem. The optimal solution is typically found at one of the vertices or corner points of this region, making it a central concept in understanding how to identify the best overall outcome.

Constraints

Constraints are the limitations or restrictions placed on the decision variables, expressed as linear inequalities or equations. They reflect the practical limits of the problem, such as production capacities, resource availability, and demand requirements, ensuring that the solution is feasible in real-world scenarios.

Decision Variables

Decision variables represent the quantities that can be controlled and optimized in a problem. In production or resource allocation problems, these variables stand for the number of units to be produced or resources to be allocated, serving as the fundamental factors whose values determine the final outcome.

Linear Programming

Linear programming is a mathematical optimization technique used to achieve the best outcome in a model whose requirements are represented by linear relationships. It is widely employed for resource allocation problems where the goal is to maximize or minimize a linear objective function subject to a set of linear constraints.

Objective Function

The objective function is the mathematical expression that needs to be optimized, typically to maximize profit or minimize cost. It represents the goal of the problem by combining the decision variables with corresponding coefficients that reflect revenue or cost contributions.

Transcript

00:01 Now over here in this question, it is being said that there are two types of notebooks.

00:07 One is the regular and the other one is the deluxe one.

00:10 We've been provided the selling cost and a production cost of each type of notebook.

00:15 The main objective of this function is we have to find out the number of each type of notebook.

00:20 That should be manufactured so that the difference between the selling price and the production cost is maximum.

00:26 So let x is equal to number of.

00:31 Of deluxe number of deluxe notebooks that is manufactured and why be the number of regular notebooks okay apart from that um like the apart from the selling price and the production cost we've been given some conditions over here it is being said that the manufacture of a deluxe notebook should be between um 2 000 and 3 000 so we are going to write the constraint equations based on the conditions given to us so since we've denoted deluxe notebooks by x so that should be it should be greater than or equal to 2000 yeah and it should be less than or equal to 3 ,000 now again for the why for why that is for the number of regular notebooks also it has been said that the production should be between 3 ,000 to 5 ,000 so i'm writing one more equation over here now i can see one more thing in the question that is the total number of book the total production of both the types of notebook should not exceed above 7 ,000 so x plus y should be less than or equal to 7000 so i've got three constraint equations over here now, let us write the objective function first of all.

02:16 So for the objective function, it has been said that we should get the maximum difference of the selling price and the production cost.

02:23 So if i just add up both the selling prices and both the production and then subtract the sum of the production cost, we will get the objective function.

02:32 So for that, i can just simply write that as 4x plus 3x, 4x plus 3 by minus 3.

02:48 2x plus 2 .6y.

02:53 Now just simplify this, just add and subtract the like terms together.

02:59 So we'll be getting p is equal to 0 .8x plus 0 .4 .y.

03:09 This is our objective function.

03:11 Now as you can see on the right side, i've already drawn the graph.

03:15 And based on this graph, we can see the visible region as this one.

03:19 See? the one that i'm trading.

03:21 This is the visible region or you can see the solution region based on the constraints...