Reference no: EM132287825
Question 1. A financial adviser recently received a call from a client who wanted to invest a portion of a $150,000 inheritance. The client wanted to realize an annual income, but also wanted to spend some of the money. After discussing the matter, the client and the adviser agreed that a mutual fund, corporate bonds, and a money market account would make suitable investments. The client was willing to leave allocation of the funds among these investment vehicles to the financial adviser, but with these provisions:
1. At least 20 percent of the amount invested should be in the money market account.
2. The investment must produce at least $12,000 annually.
3. The uninvested portion should be as large as possible.
The annual returns would be 11 percent for the mutual fund, 8 percent for the bonds, and 7 percent for the money market. Formulate an LP model that will achieve the client's requests. Ignore transaction costs, the adviser's fee, and so on. (Flint: All terms in the objective function must include a variable.)
Question 2. Wineco produces wine coolers and sells them to retail distributors. One popular blend consists of wine, apple juice, and grape juice. Operators must adhere to certain guidelines when preparing the wine cooler:
1. At least 10 percent of the mix must be grape juice.
2. The ratio of apple juice to grape juice must be 2 1/2:1.
3. The mix must contain between 20 percent and 25 percent wine.
The company pays $1.20 per gallon for wine, $1.40 per gallon for grapes juice, and Formulate an LP model that will help the owner to determine how much wine cooler to mix each day if the capacity of the mixing equipment is 200 gallon per day and the wine cooler is sold for $4 a gallon. The owner wants to maximize profits . Ignore mixing and bottling costs.
Question 3. The planning committee of a bank makes monthly decisions on the amount of funds to allocate to loans and to government securities. Some of the loans are secured backed by collateral such as a home or an automobile), and some are unsecured. A list of the various types of loans and their annual rate of return are shown in Table 4-18,
| Table 4-18 Annual Rate of Return for Investments |
| Type of Investment |
Annual Rate of Return |
| Secured loans |
|
| Residential mortgage (x1) |
11 |
| Commercial mortgage (x2) |
12 |
| Automobile (x3) |
15 |
| Home improvement (x4) |
13 |
| Unsecured loans |
|
| Vacation (x5) |
17 |
| Student (x6) |
10 |
| Government securities (x7) |
9 |
In making its decision, the planning committing must satisfy certain legal requirements and bank polices. These can be summarized by the following set of conditions:
1. The amount allocated to secured loans must be at least four times the amount allocated to unsecured loans.
2. Auto and home loans should be no more than 20 percent of unsecured loans.
3. Student loans should be no more than 30 percent of unsecured loans.
4. The amount allocated to government securities should be at least 10 percent, but no more than 20 percent, of available funds.
5. The amount allocated to vacation loans must not exceed 10 percent of all loans. The bank has $5 million available for loans and investments in the next month.
Formulate a linear programming model that will enable the Planning committee to determine the optimal allocation of funds if the objective is to maximize the annual return, given the preceding list of conditions.
Question 4. A classic linear programming problem involves minimizing trim loss. Here is one version of the problem:
A mill cuts 20-foot pieces of wood into several different lengths: 8-foot, 10-.foot, and 12-foot. The mill has a certain amount of 20-foot stock on hand and orders for the various sizes. The objective is to fill the orders with as little waste as possible. For example, if two 8-foot lengths are cut from a 20-foot piece, there will be a loss of 4 feet, the leftover amount.
Currently, the mill has 350 20-foot pieces of wood on hand and the following orders, which must be filled from stock on hand