# Using a Graphic Method to solve Linear Programming

The Electrocomp Corporation manufactures two electrical products: air conditioners and large fans. The assembly process for each is similar in that both require a certain amount of wiring and drilling. Each air conditioner takes 3 hours of wiring and 2 hours of drilling. Each fan must go through 2 hours of wiring and 1 hour of drilling. During the next production period, 240 hours of wiring time are available and up to 140 hours of drilling time may be used. Each air conditioner sold yields a profit of $25. Each fan assembled may be sold for a $15 profit. Formulate and solve this LP production mix problem to find the best combination of air conditioner and fans that yields the highest profit. Use the corner point graphical approach.

Phase I: Formulate the Linear Programming Model

Step 1: Define variables

Step 2: Write objective function

Step 3: Write constraints

Phase II: Solve the LP Model using the Graphical Method

Step 1: Plot the constraints

Step 1(a): Write all constraints as equations

Step 1(b): Determine two points for each equation

Step 1(c): Connect the points to draw the straight lines.

Step 1(d) Indicate the direction, toward or away from the origin.

Step 2: Identify the feasible region

Step 3: Identify the corner points of the feasible region

Step 4: Test the corner points to determine the optimal solution

Corner Points Total Profit

Write down the optimal numbers for variables and objectives as stated in Phase I Step 1 and 2.

Optimal Solution

Number of Air-conditioners =

Number of Fans =

Total Profit =

Serendipity*

The three princes of Serendip

Went on a little trip.

They could not carry too much weight;

More than 300 pounds made them hesitate.

They planned to the ounce. When they returned to Ceylon

They discovered that their supplies were just about gone

When, what to their joy, Prince William found

A pile of coconuts on the ground.

"Each will bring 60 rupees," said Prince Richard with a grin

As he almost tripped over a lion skin.

"Look out!" cried Prince Robert with glee

As he spied more lion skins under a tree.

"These are worth even more - 300 rupees each

If we can just carry them all down the beach.

Each skin weighed fifteen pounds and each coconut, five,

But they carried them all and made it alive.

The boat back to the island was very small

15 cubic feet capacity - that was all.

Each lion skin took up one cubic foot

While eight coconuts the same space took.

While everything was stowed they headed to the sea

And on the way calculated what their new wealth might be.

"Eureka!" cried Prince Robert, "Our wealth is so great

That there's no other way we could return in this state.

And any other skins or nut which we might have brought

Would now have us poorer. And now I know what -

I'll write my friend Horace in England, for surely

Only he can appreciate our serendipity."

Formulate and solve Serendipity by graphical linear programming to calculate "what their new

wealth might be".

Problem 1

Step 1: Define variables

L = Number of Lion Skins

C = Number of coconuts

Step 2: Write objective function

Maximize Total Wealth Z = 300 L + 60 C

Step 3: Write constraints

Weight: 15 L + 5 C ≤ 300

Space: L + (1/8) C ≤15 or 8 L + C ≤ 120

Note: The space constraint has been simplified by multiplying both sides by 8.

Non-negativity: L ≥0, C ≥0

Phase II: Solve the LP Model using the Graphical Method

Step 1: Plot the constraints

Step 1(a): Write all constraints as equations

Step 1(b): Determine two points for each equation

Step 1(c): Connect the points to draw the straight lines.

Step 1(d) Indicate the direction, toward or away from the origin.

Step 2: Identify the feasible region

Step 3: Identify the corner points of the feasible region

Step 4: Test the corner points to determine the optimal solution

Corner Points Total Wealth

( L , C ) Z = 300 L + 60 C

Write down the optimal numbers for variables and objectives as stated in Phase I

Step 1 and 2.

Optimal Solution

L = Number of Lion Skins =

C = Number of coconuts =

Total Wealth Z =

#### Solution Summary

The solution provides explanations how to use graphic method to solve linear programming problems.