Exercise 7.10

A New Hampshire produce company supplies organically grown

apples to four New England specialty stores. After the apples are

collected at the company's orchard, they are transported to any of

three preparation centers where they are prepared for retail (by

undergoing extensive cleaning and then packaging), after which the

prepared apples are shipped to the specialty stores. The cost of

transporting organic fruit is rather high due to the fragility of the

product.

The company has three preparation centers available for use. Table

7.43 shows the unit shipping and preparation costs (in $/lb.) from the

company's orchard to the preparation centers, as well as the

preparation centers' monthly capacities. Demand at the specialty

stores is shown in Table 7.44. The unit costs of transporting the apples

from the preparation centers to the specialty stores is shown in Table

7.45.

TABLE 7.43 Unit costs and monthly capacities at the three preparation centers.

PreparationCenter / Unit Transportation and Preparation Cost ($/lb) / Monthly Capacity (lbs.)

1 $0.30 200

2 $0.60 450

3 $0.90 760

TABLE 7.44 Monthly demand for apples at four specialty stores.

Store Monthly Demand (lbs.)

Organic Orchard 300

Fresh & Local 500

Healthy Pantry 400

Season's Harvest 200

TABLE 7.45 Unit transportation costs for transporting prepared apples from the preparation centers to the specialty stores, in $/lb.

Preparation Center / Organic Orchard / Fresh & Local / Healthy Pantry / Season's Harvest

1 $0.40 $0.60 $0.30 $0.70

2 $0.60 $0.60 $0.30 $0.70

3 $0.10 $0.70 $0.60 $0.90

The produce company must decide the amount of apples to prepare at

each of the three preparation centers and how much of each specialty

store's demand to supply from each preparation center, at least cost.

(a) Construct a linear optimization model to determine the amount

of apples to be prepared at each of the three preparation centers,

and how much of each specialty store's demand to supply from

each preparation center, in order to minimize the total cost of the

operation.

Hint: Notice that the produce company must decide both the

amount of apples to prepare at each preparation center and how

many apples to ship from each preparation center to each store. Let

X1O, for example, be the decision variable defined to be the

quantity of apples prepared at preparation center 1 and shipped to

the Organic Orchard store. Let X1F, X1H, and X1S be defined

likewise for the other three specialty stores, and define the

appropriate decision variables for the other two preparation centers

in a similar manner. Then the constraints in the problem should

include preparation center capacity constraints, such as:

X1O + X1F + X1H + X1S 200,

and store demand constraints, such as:

X1O + X2O + X3O 300,

for example.

(b) Solve the linear optimization model on the computer. What

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