Solving LP with isoprofit line method

Summary

In this episode, Yen-Ting Lin explains the isoprofit line method to solve linear programming (LP) problems. Isoprofit lines are designed to graphically determine the optimal solution within a feasible region, highlighting which point maximizes the objective function. The essence of the method lies in drawing an isoprofit line, assigning a value, and adjusting this value to explore possible solutions. The line is moved to tangentially touch a feasible region corner point, revealing the optimal solution lying at this intersection. This process reinforces the critical role of geometry in solving LP problems and helps identify at least one corner point contributing to the optimal solution.

Highlights

• Graphically solve LP by first plotting the feasible region! 📊
• Assign a value to the objective function and draw an isoprofit line. ✏️
• Increase the objective function value and move the isoprofit line to find better options! 🚀
• Optimal solutions often lie at the intersection with feasible region's corner points! 🎯

Key Takeaways

• The isoprofit line method graphically solves LP problems by moving a line within a feasible region! 📈
• Geometry is your friend! The optimal solution intersects a corner point of the feasible region. 🌟
• Increasing the objective function value moves the isoprofit line, revealing better solutions. 🎯

Overview

Yen-Ting Lin walks us through the nifty isoprofit line method to tackle linear programming challenges. His step-by-step approach demystifies graphical solutions by initially plotting the feasible region and then strategically moving an isoprofit line to pinpoint the optimal solution. It's an engaging exploration of how a line can navigate a complex geometric space and find that golden intersection yielding the best outcomes!

The beauty of the isoprofit line method lies in its simplicity. By assigning and adjusting values, the method moves the line to discover where it intersects with the feasible region's boundaries, ultimately revealing the optimal value. The process stands as a testament to the power of visualization in solving mathematical puzzles and sheds light on the rewarding intersection of mathematics and art.

Lin's explanation not only teaches us to optimize through simple geometry but also emphasizes the vital lesson that in LP problems, the magic often happens at the corners. It's a clever dance of lines and points where each move toward a higher value is meticulously designed to identify that one elusive point where maximum value is achieved.

Chapters

• 00:00 - 00:30: Introduction to ISO Profit Line Method This chapter introduces the ISO Profit Line method, a solution technique for linear programming problems. It starts by describing a linear programming problem and proceeds to explain how to solve it graphically. The graphical method involves plotting the feasible region, which includes all points that satisfy the given constraints. This foundational step is crucial for understanding and applying the ISO Profit Line in solving such problems.
• 00:30 - 01:00: Steps in ISO Profit Line Method The Steps in ISO Profit Line Method chapter focuses on maximizing an objective function within a feasible region. It outlines a three-step approach to the isoprofit line method, beginning with assigning an appropriate value to the objective function.
• 01:00 - 01:30: Assigning Values to Objective Function This chapter discusses how to assign values to an objective function and graphically represent it. The method involves selecting a particular value, such as 2100, and then determining the line it forms on a graph. The speaker explains a technique to graph this by first finding the intercepts on the axes, such as setting t equal to 0, resulting in C equaling 42. This intercept is then used to plot the corresponding line on the graph, indicating a practical step in visualizing objective functions.
• 01:30 - 02:00: Drawing the ISO Profit Line The chapter titled 'Drawing the ISO Profit Line' begins with the explanation of how to determine points on a specific line. It demonstrates setting one variable, C, to zero to solve for T, resulting in the coordinates (30, 0) as a point on the line. By identifying two points, the chapter describes connecting them to form a line, which is referred to as an ISO profit line. The significance of this line is that every point on it will yield the same level of profit, hence the name 'ISO profit line'.
• 02:00 - 02:30: Improving the Objective Value The chapter titled 'Improving the Objective Value' discusses the process of optimizing a numerical objective value. Initially, a value of 2100 is considered but is not deemed optimal. By adjusting and increasing this number, a better solution is found with a new value of 2800. The chapter likely explores graphing techniques to illustrate this improvement in the objective value.
• 02:30 - 03:00: Finding the Optimal Solution In the chapter titled 'Finding the Optimal Solution,' the discussion explores the process of intersecting with the feasible region in order to improve the objective. The speaker begins by noting how the objective can be improved from 2,100 to 2,800 and considers whether further improvements are possible. By testing with a larger number, such as 3,500, they observe that it's indeed feasible to continue increasing this number. It's evident that as the number increases, the isoprofit line shifts upwards, demonstrating the potential for continuous improvement.
• 03:00 - 03:30: Recap of ISO Profit Line Method The chapter explains the concept of determining the optimal solution using the ISO Profit Line Method in linear programming. The optimal solution is identified at the intersection of the line with the feasible region where the objective function can no longer be improved. The process concludes when the profit line tangentially intersects with the feasible region at that singular point, marking it as the optimal solution.
• 03:30 - 04:00: Purpose of ISO Profit Line Method The chapter discusses the ISO Profit Line Method and its purpose. It explains how to find the optimal solution by solving two equations with two unknowns, which in this case yields the point (30, 40). The chapter concludes with a recap that the ISO Profit Line Method involves drawing an isoprofit line and moving it in the direction that enhances the objective.
• 04:00 - 05:00: Characteristics of Optimal Solution The chapter discusses the characteristics of an optimal solution in the context of geometry. It highlights that such a solution typically occurs where the isoprofit line intersects with a corner point of the feasible region. The chapter also explains that the purpose of drawing the isoprofit line is to determine its slope to understand how steep it is.

Solving LP with isoprofit line method Transcription

• 00:00 - 00:30 hi I'm going to explain what is so-called an ISO profit line solution method for a linear programming problem let's say we have this linear programming problem and we're going to solve for it how can we start well we can graphically solve it by first graphing the so-called feasible region and here it is so feasible region is the region in which all the points satisfy all of the constraints with that our
• 00:30 - 01:00 goal is basically to find within this feasible region which point point which point will maximize this objective function isoprofit line method involves three steps I'm going to explain those three steps one by one in the following slide to start off I'm going to assign an appreciative value to the objective
• 01:00 - 01:30 function let's say I choose 2100 so that means the objective function looks like this so it's a line I can draw this line on the graph so let's do it there are many ways to draw a line on the graph one of the ways that the way I choose is to find the intercept to the axis first I set t equal to 0 which means that C will be 42 from solving this equation which means that well 0 42
• 01:30 - 02:00 is a point on this line similarly I can set C to 0 solving this equation I get T being 30 which means that 30 0 is another point on this line here are these two points I can connect them and that is this function we call this line an ISO profit line because every point on this line will give you the same
• 02:00 - 02:30 objective value 2100 in this case okay well is 2100 the best we can get apparently not let's try another number let's increase this number a little bit looks like 2,800 and I graph this line here apparently 2,800 is a better solution better objective than 2,100 and we still
• 02:30 - 03:00 have and he still intersect with the feasible region so we can improve our objective from 2,100 to 2800 then can we do do any way better let's try another larger number 3,500 well it looks like this so yes we can keep increasing this number and we observe that as we increase this number this ipro isoprofit line moves up so we can keep doing that
• 03:00 - 03:30 all the way until like this until the line intersect exactly at one point with the feasible region at this point we cannot further improve or increase the value okay so what is this point this is our optimal solution thing so it's the intersection of these two blue lines so
• 03:30 - 04:00 we simply can find this point by solving these two equations with two unknowns turns out to be this point 30 40 so this is our optimal solution so let me recap what is essentially the isoprofit line method it has a very simple concept so we started off by drawing on isoprofit line and then move it toward the direction that improves the objective
• 04:00 - 04:30 value by the geometry we know that the optimal solution will occur at the situation where the isoprofit line intersects with at least one of the corner points of the feasible region so what is the purpose of drawing the isoprofit line essentially the purpose is to find the slope how steep this line is
• 04:30 - 05:00 so that we can move around and find out where does it intersect with the corner points of the feasible region because that corner point will be our optimal solution furthermore the geometry I just show you also reveals a very important characteristics of this optimal solution the optimal solution of an LP problem definitely contains at least one of the
• 05:00 - 05:30 corner points of the feasible region