Note: Please solve this question manually using microsoft word. Do not use excel solver to solve it

Write out the decision variables, constraints and objective function

The Darby Company manufactures and distributes meters 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 Fort Worth, Texas, and later, as business expanded, 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 companys production plants. The cost of each meter produced at the El Paso plant is $10.50. The San Bernardino plant utilizes newer and more efficient equipment; as a result, manufacturing cost is $0.50 per meter less than at the El Paso plant.

Due to the companys rapid growth, not much attention had been paid to the efficiency of its supply chain, but Darbys management decided that it is 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 6.10.

Table 6.10

Shipping Cost per Unit from Production Plants to Distribution Centers (In $)

	Distribution Center
Plant	Fort Worth	Santa 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 El Paso plant and 20,000 meters at the San Bernardino plant. Note that no shipments are allowed from the San Bernardino plant to the Fort 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 6.11.

Table 6.11

Quarterly Demand Forecast

Customer Zone	Demand (meters)
Dallas	6300
San Antonio	4880
Wichita	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 6.12; note that some distribution centers cannot serve certain customer zones. These are indicated by a dash, .

Table 6.12

Shipping Cost from the Distribution Centers to the Customer Zones

					Customer Zone
Distribution Center	Dallas	San Antonio	Wichita	Kansas City	Denver	Salt Lake City	Phoenix	Los Angeles	San Diego
Fort Worth	0.3	2.1	3.1	4.4	6.0
Santa 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	2.4	2.1	2.5

In its current supply chain, demand at the Dallas, San Antonio, Wichita, and Kansas City customer zones is satisfied by shipments from the Fort Worth distribution center. In a similar manner, the Denver, Salt Lake City, and Phoenix customer zones are served by the Santa Fe distribution center, 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.

1. If the company does not change its current supply chain, what will its distribution costs be for the following quarter?

2. Suppose that the company is willing to consider dropping the distribution center limitations; that is, customers could be served by any of the distribution centers for which costs are available. Can costs be reduced? If so, by how much?

3. The company wants to explore the possibility of satisfying some of the customer demand directly from the production plants. In particular, the shipping cost is $0.30 per unit from San Bernardino to Los Angeles and $0.70 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-to-customer shipments?

4. Over the next five years, Darby is anticipating moderate growth (5000 meters) to the north and west. Would you recommend that Darby consider plant expansion at this time? Please solve this question manually on microsoft word. Do not use excel to solve it