Transportation Problem

The Darby Company manufactures and distributes meter used to measure electric power consumption. The company started with a small production plant in El Paso and gradually built a customer base throughout Texas.
A distribution center was established in Ft. Worth, Texas later, as business expanded to the north, a second distribution center was established in Santa Fe, New Mexico.
The El Paso plant was expanded when the company began marketing
its meters in Arizona, California, Nevada, and Utah. With the growth of the West Coast business, the Darby Company opened a third distribution center in Las Vegas and just two years ago opened a second production plant in San Bernardino, California.
Manufacturing costs differ between the company's production plants.
The cost of each meter produced at the El Paso plant is $10.50. The
San Bernardino plant utilizes new and more efficient; as a result,
manufacturing costs are $.50 per meter less than at the El Paso plant.
The company's rapid growth meant that not much attention was paid to
the efficiency of the distribution system. Darby's management decided
it is now time to address this issue. The cost of shipping a meter
from each of the two plants to each of the three distribution centers is shown in Table 1 below.

Table 1
Shipping cost per unit ($) from prod. plants to distribution centers

Distribution Center
Plant Ft Worth Sante Fe Las Vegas
El Paso 3.20 2.20 4.20
San Bernardino --- 3.90 1.20

The quarterly production capacity is 30,000 meters at the older ElPaso
plant and 20,000 meters at the San Bernardino plant. Note that no
shipments are allowed from the San Bernardino plant to the Ft. Worth
distribution center.
The company serves nine customer zones from the three distribution
centers. The forecast of the number of meters needed in each customer zone for the next quarter is shown in Table 2.

Table 2: Quarterly Demand Forecast
Customer Zone Demand (meters)
Dallas 6300
San Antonio 4880
Wichitia 2130
Kansas City 1210
Denver 6120
Salt Lake City 4830
Phoenix 2750
Los Angeles 8580
San Diego 4460

The cost per unit of shipping from each distribution center to each
customer zone is given in Table 3. Note that some of the distribution
centers cannot serve certain customer zones.

Table 3: Shipping cost from distribution centers to customer zones ($)
Customer Zone
San Kansas Salt Lake
Dallas Antonio Wichita City Denver City Phoenix LA SD
Ft Wor 0.3 2.1 3.1 4.4 6.0 -- -- -- --
Sante Fe 5.2 5.4 4.5 6.0 2.7 4.7 3.4 3.3 2.7
Las Vegas -- -- -- -- 5.4 3.3 2. 2.1 2.5

In the current distribution system, demand at the Dallas, San Antonio,
Wichitia, and Kansas City customer zones is satisfied by shipments from
the Ft. Worth distribution center. In a similar manner, the Denver,
Salt Lake City, and Phoenix customer zones are served by the Santa Fe
distribution, and the Los Angeles and San Diego customer zones are served by the Las Vegas distribution center. To determine how many units to ship from each plant, the quarterly customer demand forecasts are aggregated at the distribution centers and a transportation model is used to minimize the cost of shipping from the production plants to the distribution centers.

Question for overall problem
1. If the company does not change its current distribution strategy,
what will its manufacturing and distribution costs be for the following
quarter?
2. Suppose that the company is willing to consider dropping the
distribution center limitations; that is, customer could be served by any of the distribution centers for which costs are available. Can costs be reduced? By how much?
3. The company wants to explore the possibility of satisfying some for
the customer demand directly from the production plants. In
particular, the shipping cost is $.30 per unit from San Bernardino to Los Angeles and $.70 from San Bernardino to San Diego. The cost for direct shipments from El Paso to San Antonio is $3.50 per unit. Can distribution costs be further reduced by considering these direct plant customer shipments?
4. Over the next five years. Darby is anticipating moderate growth
(5000 meters) to the North and West. Would you recommend that they
consider plant expansion at this time?