# Linear Programming and Maximization

• rams_man13
In summary, the conversation discusses the production process at AutoIgnite, where two components are produced at plants in Buffalo and Dayton and then sent to the Cleveland plant for assembly. The goal is to formulate a linear programming model to maximize daily production of ignition systems at Cleveland. The optimal solution is for Buffalo to use 80% of their time producing component 1 and 20% of their time producing component 2, while Dayton should use 100% of their time producing component 2. These values can also be represented by the x-intercept on a graph where the lines for component 1 and component 2 production intersect.
rams_man13

## Homework Statement

AutoIgnite produces electronic ignition systems for automobiles at a plant in Cleveland, Ohio. Each ignition system is assembled from two components produced at AutoIgnite’s plants in Buffalo, New York, and Dayton, Ohio. The Buffalo plant can produce 2000 units of component 1, 1000 units of component 2, or any combination of the two components each day. For instance, 60% of Buffalo’s production time could be used to produce component 1 and 40% of Buffalo’s production time could be used to produce component 2; in this case, the Buffalo plant would be able to produce 0.6(2000) = 1200 units of component 1 each day and 0.4(1000) = 400 units of component 2 each day. The Dayton plant can produce 600 units of component 1, 1400 units of component 2, or any combination of the two components each day. At the end of each day, the component production at Buffalo and Dayton is sent to Cleveland for assembly of the ignition systems on the following work day.

a. Formulate a linear programming model that can be used to develop a daily production schedule for the Buffalo and Dayton plants that will maximize daily production of ignition systems at Cleveland.

b. Find the optimal solution.

## Homework Equations

I'm not sure what this means exactly.

## The Attempt at a Solution

Let x equal % of time producing C1 in B
Let y equal % of time producing C1 in D
Let 1-x equal % of time producing C2 in B
Let 1-y equal % of time producing C2 in D

C1 produced: 2000x+600y <= 0
C2 produced: 1000(1-x)+1400(1-y)<=0

Iginitions are made of a C1 and a C2. Therefore, set the equations equal to each other?

2000x+600y=1000(1-x)+1400(1-y)
3000x+2000y= 2400

0<=x,y <=1

Then I graph the line of where C1 and C2 are equal and the line of C1 and the line of C2.

I take the coordinates where C1 crosses with the equality line and where c2 crosses with the equality line. But when I plug those into the equation, I get 2400 for both, which leads me to believe it's wrong.

Not sure if this is right, but i took the line that is plotted from
2000x+600y=1000(1-x)+1400(1-y),
found the x and y intercepts (where all production capacity is dedicated to one component)
and then found the midpoint between them (where it would be evenly split)
got x=2/5, y=3/5, and total production is 1160 for each.
is this right?

| Y | C1 | C2
-----------------------------------------------------------------------------
X | Y= 1 1/5 - 1 1/2X | 2000X + 600Y | 1000(1-X) + 1400 (1-Y)
-----------------------------------------------------------------------------
0 | 1.2 | 720 | 720
------------------------------------------------------------------------------
.1| 1.05 | 830 | 830
------------------------------------------------------------------------------
.2| 0.9 | 940 | 940
------------------------------------------------------------------------------
.3| 0.75 | 1050 | 1050
------------------------------------------------------------------------------
.4| 0.60 | 1160 | 1160
------------------------------------------------------------------------------
.5| 0.45 | 1270 | 1270
------------------------------------------------------------------------------
.6| 0.30 | 1380 | 1380
------------------------------------------------------------------------------
.7| 0.15 | 1490 | 1490
------------------------------------------------------------------------------
.8| 0.0 | 1600 | 1600
------------------------------------------------------------------------------

This is some of the help I have received by using a t-table. This is the right answer, I just need it in inequalities and graphical form.

The answer is Buffalo should use 80% of their time producing C1 and 20% of their time producing C2. Dayton should use 100% of their time producing C2.

that is the same as the x intercept. is that coincidence, or is that the case all of the time?

## 1. What is linear programming?

Linear programming is a mathematical method used for finding the best possible solution to a problem that involves constraints and multiple variables. It involves maximizing or minimizing an objective function while satisfying a set of linear constraints.

## 2. How is linear programming used in real life?

Linear programming is widely used in industries such as finance, transportation, manufacturing, and telecommunications. It can be used to optimize production schedules, minimize costs, and allocate resources efficiently.

## 3. What is the difference between linear programming and linear optimization?

Linear programming and linear optimization are often used interchangeably. However, linear optimization is a broader term that refers to the process of maximizing or minimizing an objective function while satisfying constraints, which may not necessarily be linear. Linear programming specifically deals with linear constraints and objectives.

## 4. Can linear programming be used for non-linear problems?

No, linear programming can only be used for problems with linear constraints and objectives. For non-linear problems, other optimization methods such as quadratic programming or non-linear programming would be more suitable.

## 5. How do you solve a linear programming problem?

Linear programming problems can be solved using various techniques such as the simplex method, the interior-point method, or the graphical method. These methods involve converting the problem into a standard form, finding the optimal solution, and then interpreting the results in the context of the problem.

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