The famous Y.S. Chang Restaurant is open 24 hours a day. Waiters report for duty at 3 A.M., 7 A.M., 11 A.M., 3 P.M., 7 P.M., or 11 P.M., and each works an 8-hour shift. Shifts pay different rates. See Table A for the shifts and their rates. Table B shows the minimum number of workers needed during the six periods into which the day is divided. Chang’s scheduling problem is to determine how many waiters should report for work at the start of each time period to minimize the total cost of staff required for one day’s operation. Solve this so that your answers are in the green boxes.

The Battery Park Stable feeds and houses the horses used to pull tourist-filled carriages through the streets of Charleston’s historic waterfront area. The stable owner, an ex-racehorse trainer, recognizes the need to set a nutritional diet for the horses in his care. At the same time, he would like to keep the overall duty cost of feed to a minimum.

The feed mixes available for the horses’ diet are an oat product, a highly enriched grain, and a mineral product. Each of these mixes contains a certain amount of five ingredients needed daily to keep the average horse healthy. The table on this page shows these minimum requirements, units of each ingredient per pound of feed mix, and costs for the three mixes.

In addition, the stable owner is aware that an overfed horse is a sluggish worker. Consequently, he determines that 6 pounds of feed per day are the most that any horse needs to function properly. Formulate this problem and solve for the optimal daily mix of the three feeds that minimizes the total cost of the mix.

Eddie Kelly is running for reelection as mayor of a small town in Alabama. Jessica Martinez, Kelly’s campaign manager during this election, is planning the marketing campaign, and there is some stiff competition. Martinez has selected four ways to advertise: television ads, radio ads, billboards, and newspaper ads. The costs of these and the audience reached by each type of ad is shown in the table below.

In addition, Martinez has decided that there should be at least six ads on TV or radio or some combination of those two. The amount spent on billboards and newspapers together must not exceed the amount spent on TV ads. Each type of ad may not be used more than 10 times. The monthly budget for advertising has been set at $15,000. How many ads of each type should be placed to maximize the total number of people reached? (Do not round)

Daniel Grady is the financial advisor for a number of professional athletes. An analysis of the long-term goals for many of these athletes has resulted in a recommendation to purchase stocks with some of their income that is set aside for investments. Five stocks have been identified as having very favorable expectations for future performance. Although the expected return is important in these investments, the risk, as measured by the beta of the stock, is also important. (A high value of beta indicates that the stock has a relatively high risk.) The expected return and the betas for five stocks are given below.

Daniel would like to minimize the beta of the stock portfolio (calculated using the sum product of each stock’s beta and the proportion of the investment in that stock). He requires an expected return of at least 11%. In addition, Daniel has decided that no more than 35% of the portfolio should be invested in any one stock.

Determine the proportion of the portfolio that should be in each stock. (For example, if 20% of the portfolio is in stock A, you will get an answer of 0.2). Since you are working with proportions, you will need a constraint that the sum of the proportions is equal to 1.