In Southern Region there are a group of three Zones (communal farming communities). Overall planning for this group is done in its Coordinating Technical Office. This office currently is planning agricultural production for the coming year. The agricultural output of each zone is limited by both the amount of available irrigable land and the quantity of water allocated for irrigation by the Water Commissioner (a national government official). These data are given in the following table.
TABLE 1 Resource data for the Southern Confederation of the Three Zones
|
Zone
|
Usable Land (Acres)
|
Water Allocation (Acre Feet)
|
|
1
|
400
|
600
|
|
2
|
600
|
800
|
|
3
|
300
|
375
|
The crops suited for this region include sugar beets, cotton, and sorghum, and these are the three being considered for the upcoming season. These crops differ primarily in their expected net return per acre and their consumption of water. In addition, the Ministry of Agriculture has set a maximum quota for the total acreage that can be devoted to each of these crops by the Southern Region of Three Zones, as shown in Table 2 below.
TABLE 2. Crop data for the Southern region of Three Zones
|
Crop
|
Maximum Water Consumption
|
Net Return Crop Quota (Acres)
|
(Acre Feet/Acre) ($/Acre)
|
|
Sugar beets
|
600
|
3
|
1,000
|
|
Cotton
|
500
|
2
|
750
|
|
Sorghum
|
325
|
1
|
250
|
Because of the limited water available for irrigation, the Southern region of three zones will not be able to use all its irrigable land for planting crops in the upcoming season. To ensure equity between the three zones, it has been agreed that every zone will plant the same proportion of its available irrigable land. For example, if zone 1 plants 200 of its available 400 acres, then zone 2 must plant 300 of its 600 acres, while zone 3 plants 150 acres of its 300 acres. However, any combination of the crops may be grown at any of the zones. The job facing the Coordinating Technical Office is to plan how many acres to devote to each crop at the respective zone while satisfying the given restrictions. The objective is to maximize the total net return to the Southern region of three zones as a whole.
Investment Problem
3. Innis consulting group Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client's needs. For a new client, Innis has been authorized to invest up to $1.2 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of $5 per dollar invested ; each unit of the money market fund costs $100 and provides an annual rate of return of $4 for each dollar invested . The client wants to minimize risk subject to the requirement that the annual income from the investment be at least $60,000. According to Innis's risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3; the higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis's client also specified that at least $300,000 be invested in the money market fund.
a. Determine how many units of each fund Innis should purchase for the client to minimize the total risk index for the portfolio.
b. How much annual income will this investment strategy generate?
c. Suppose the client desires to maximize annual return. How should the funds be invested?
d. What is the optimal solution, and what is the minimum total risk?
e. Specify the objective coefficient ranges.
f. How much annual income will be earned by the portfolio?
g. What is the rate of return for the portfolio?
h. What is the dual value for the funds available constraint?
i. What is the marginal rate of return on extra funds added to the portfolio?
j. Suppose the risk index for the stock fund (the value of CS) increases from its current value of 8 to 12. How does the optimal solution change, if at all?
k. Suppose the risk index for the money market fund (the value of CM) increases from its current value of 3 to 3.5. How does the optimal solution change, if at all?
l. Suppose CS increases to 12 and CM increases to 3.5. How does the optimal solution change, if at all?