3. A publisher has orders for 600 copies of a certain text from San Francisco and 400 copies from Sacramento. The company has 700 copies in a warehouse in Novato and 800 copies in a warehouse in Lodi. It costs $5 to ship a text from Novato to San Francisco, but it costs $10 to ship it to Sacramento. It costs $15 to ship a text from Lodi to San Francisco, but it costs $4 to ship it from Lodi to Sacramento. How many copies should the company ship from each warehouse to San Francisco and Sacramento to fill the order at the least cost?

Steps

0. Read the whole problem.

1. Define your unknowns.

2. Express the objective function

3. Express
the constraints.

4. Graph the constraints.

5. Find the cornerpoints to the region of feasible solutions.

6. Evaluate the objective function at all the feasible corner points.

After reading the whole problem we see that they want to know how many books to ship from each warehouse to each bookstore. There are four unknowns

Let

x = the number of books from Novato to San Francisco

y = the number of books from Novato to Sacramento

z = the number of books from Lodi to San Francisco

w = the number of books from Lodi to Sacramento.

Thge objective is to minimize the cost

cost = 5x + 10y + 15z + 4w

The first two constraints have to do with the orders.

San Francisco

x + z = 600

Sacramento

y + w = 400

These are equations. San Francisco has an order for 600 books. They have to get exactly 600 books. More or less will not do. If they get less than 600 books, students will be going without texts, but if they get more than 600 texts, they will say, "We only orderd 600 texts. We're not paying for any more." Similarly, Sacramento needs exactly 400 texts. As a result, we can solve these equations to express z and w in terms of x and y.

z = 600 - x

w = 400 - y

The next two cnstraints have to do with the supplies

There are only 700 books in Novato.

x + y __<__ 700

There are only 800 books in Lodi.

z + w __<__ 800

If we substitute for z and w, we can express this constraint in terms of x and y.

600 - x + 400 - y __<__ 800

1000 - x - y __<__ 800

200 __<__ x + y

__>__ 0

y
__>__ 0

z __>__ 0

w __>__ 0

When we substitute into the last two, we get

600 - x __>__ 0

400 - y __>__ 0

or

600 __>__ x

400 __>__ y

Let us summarize our constraints expressed using only x and y.

x + y __<__ 700

x + y __>__ 200

x
__<__ 600

y __<__ 400

__>__ 0

y __>__ 0

(0, 200)

(200, 0)

(600, 0)

(600, 100)

((300, 400)

We now evaluate the objective function at all of the feasible corner points. The coordinates tell us how many books are being shipped from Novato to San Francisco and Sacramento. Once we know that we can figure how many books are going to be shipped from Lodi. We will ship enough books from Lodi to fill the orders.

At (0, 400) we would be shipping 400 books from Novato to Sacramento. That would fill the Sacramento order, so we would not need to ship any books from Lodi to Sacramento, but San Francisco still needs its 500 copies. Those would all come from Lodi. This would probably be the worst solution. All of the books would be shipped to the most expensive places.

At (0, 200), 200 copies would go from Novato to Sacramento, and none from Novato to San Francisco. All 600 copies to San Francisco would come from Lodi. Sacramento would still need 200 copies which would also come from Lodi. This is a feasible solution, because there would be 800 copies coming from Lodi, and there are 800 copies in Lodi. This would be a better solution than the first one because not all copies are going to the most expensive places. The constraint upon which we find these two solutions is the one where there are no copies going from Novato to San Francisco.

At (200, 0),

(0, 200)

= 10600 - 10(0) + 6(200) = 10600 - 0 + 1200

= $11800

(200, 0)

= 10600 - 10(200) + 0 = 10600 - 2000 + 0

= $8600

(600, 0)

= 10600 - 10(600) + 6(0) = 10600 - 6000 + 0

= $4600

(600, 100)

=10600 - 10(600) + 6(100) = 10600 - 6000 + 600

= $5200

(300, 400)

= 10600 - 10(300) + 6(400) = 10600 - 3000 + 2400

= $10000

The least cost of $4600 is found at (600, 0). This makes sense if you think about it. You are fillling the entire San Francisco order with copies from Novato, which is cheaper, and filling the entire Sacramento order with copies from Lodi, which is also cheaper. One could arrive at this solution by common sense, but it is good if our techniques validate common sense.