1. The GoGo Bunny is a hot toy this Christmas, and the manufacturer has decided to ration supply to all retailers. A large retail chain owns two channels-a discount channel and a high-service channel. The retailer plans to sell the toy at a margin of $4 in the discount channel and a margin of $8 in the high-service channel. The manufacturer sends 100,000 GoGo Bunnies to the retailer. The retailer has forecast that the demand for the toy at the high-service channel is normally distributed, with a mean of 400,000 and a standard-deviation of 150,000. How many toys should the retailer send to the high-service channel?
2. A trucking firm has a current capacity of 200.000 cubic feet. A large manufacturer is willing to purchase the entire capacity at $0.10 per cubic foot per day. The manager at the trucking firm has observed that on the spot market, trucking capacity sells for an average of $0.13 per cubic foot per day. Demand, however, is not guaranteed at this price. The manager forecasts daily demand on the spot market to be normally distributed, with a mean of 60,000 cubic feet and a standard devation of 20,000. How much trucking capacity should the manger save for the spot market?
3. A department store has purchased 5.000 swimsuits to be sold during the summer sales season. The season lasts three months. and the store manager forecasts that customers buying early in the season are likely to be less price sensitive and those buying later in the season are likely to be more price sensitive. The demand curves in each of the three months are forecast to be as follows: d1 = 2,000 - 10 p1/ , d2 = 2,000(X) - 20p2, and d3 = 2,000 - 30p3. If the department store is to charge a fixed price over the entire season, what should it be? What is the resulting revenue? If the department store wants dynamic prices that vary by month, what should they be? How does this affect profits relative to charging fixed prices? If each swimsuit costs $40 and the store plans to charge dynamic prices, how many swimsuits should it purchase at the beginning of the season?
4. Greg Pickett, coach of a little-league baseball team, is preparing for a series of four games against four opponents. Greg would like to increase the probability of winning as many as games as possible by carefully scheduling his pitchers against the teams they are each most likely to defeat. Because the games are to be played back to back in less than one week, Greg cannot count on any pitcher to start in more than one game.
Greg knows the strengths and weaknesses not only of his pitchers but also of his opponents, and he believes he can estimate the probability of winning each of the four games with each of the four starting pitchers. Those probabilities are listed in the table at the top of the next column.
| Starting Pitcher |
Opponent |
| Des Moines |
Davenport |
Omaha |
Peoria |
| Johns |
0.4 |
0.8 |
0.5 |
0.6 |
| Baker |
0.3 |
0.4 |
0.8 |
0.7 |
| Parker |
0.8 |
0.8 |
0.7 |
0.9 |
| Wilson |
0.2 |
0.3 |
0.4 |
0.5 |
What pitching rotation should Greg set to provide the highest sum of the probabilities of winning each game for this team? Please formulate as a linear program.