Linear Programming and Maximization

[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.

 

[soandos]

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?

 

[rams_man13]

| 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.

 

[soandos]

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