Problem: Case: Metropolitan Police Patrol.

The Metropolitan Police Department had recently been criticized in the local media for not responding to police calls in the downtown area rapidly enough. In several recent cases, alarms had sounded for break-ins, but by the time the police car arrived, the perpetrators had left and in one instance a store owner had been shot. Sergeant Joe Davis had been assigned by the chief as head of a task force to find a way to determine optimal patrol area (dimensions) for their cars that would minimize the average time it took to respond to a call in the downtown area.

Sergeant Davis solicited help from Angela Maris, an analyst in the operations area for the police department. Together they began to work through the problem.

Joe noted to Angela that normal patrol sectors are laid out in rectangles, with each rectangle including a number of city blocks. For illustrative purposes he defined the dimensions of the sector as $x$ in the horizontal direction and as $y$ in the vertical direction. He explained to Angela that cars traveled in straight lines either horizontally or vertically and turned at right angles. Travel in a horizontal direction must be accompanied by travel in a vertical direction, and the total distance traveled is the sum of the horizontal and vertical segments. He further noted that past research on police patrolling in urban areas had shown that the average distance traveled by a patrol car responding to a call in either direction was one-third of the dimensions of the sector, or \(x / 3\) and \(y / 3\).

He also explained that the travel time it took to respond to a call (assuming a car left immediately upon receiving the call) is simply the average distance traveled divided by the average travel speed. Angela told Joe that now that she understood how average travel time to a call was determined, she could see that it was closely related to the size of the patrol area. She asked Joe if there were any restrictions on the size of the area sectors that cars patrolled, He responded that for their city, the department believed that the perimeter of a patrol sector should not be less than 5 miles or exceed 12 miles. He noted several policy issues and manpower constraints that required these specifications, Angela wanted to know if any additional restrictions existed, and Joe indicated that the distance in the vertical direction must be at least $50 \%$ more than the horizontal distance for the sector. He explained that laying out sectors in that manner meant that the patrol areas would have a greater tendency to overlap different residential, income, and retail areas than if they ran the other way. He said that these areas were layered from north to south in the city. So if a sector area were laid out east to west, all of it would tend to be in one demographic layer.

Angela indicated that she had almost enough information to develop a model, except that she also needed to know the average travel speed the patrol cars could travel. Joe told her that cars moving vertically traveled an average of 15 miles per hour, whereas cars traveled horizontally an average of 20 miles per hour. He said that the difference was due to different traffic flows.

Develop a linear programming model for this problem, and solve it using the graphical method.

Problem: Case: "The Possibility" Restaurant.

Angela Fox and Zooey Caulfield were food and nutrition majors at State University, as well as close friends and roommates. Upon graduation Angela and Zooey decided to open a French restaurant in Draperton, the small town where the university was located. There were no other French restaurants in Draperton, and the possibility of doing something new and somewhat risky intrigued the two friends. They purchased an old Victorian home just off Main Street for their new restaurant, which they named "The Possibility"

Angela and Zooey knew in advance that at least initially they could not offer a full, varied menu of dishes. They had no idea what their local customers tastes in French cuisine would be, so they decided to serve only two full-course meals each night, one

with beef and the other with fish. Their chef, Pierre, was confident he could make each dish so exciting and unique that two meals would be sufficient, at least until they could assess which menu items were most popular. Pierre indicated that with each meal he could experiment with different appetizers, soups, salads, vegetable dishes, and desserts until they were able to identify a full selection of menu items.

The next problem for Angela and Zooey was to determine how many meals to prepare for each night so they could shop for ingredients and set up the work schedule. They could not afford too much waste. They estimated that they would sell a maximum of 60 meals each night. Each fish dinner, including all accompaniments, requires 15 minutes to prepare, and each beef dinner takes twice as long. There is a total of 20 hours of kitchen staff labor available each day. Angela and Zooey believe that because of the health consciousness of their potential clientele they will sell at least three fish dinners for every two beef dinners. However, they also believe that at least \(10 \%\) of their customers will order beef dinners. The profit from each fish dinner will be approximately $12, and the profit from a beef dinner will be about $16.

Formulate a linear programming model for Angela and Zooey that will help them estimate the number of meals they should prepare each night and solve this model graphically.

If Angela and Zooey increased the menu price on the fish dinner so that the profit for both dinners was the same, what effect would that have on their solution? Suppose Angela and Zooey reconsidered the demand for beef dinners and decided that at least \(20 \%\) of their customers would purchase beef dinners. What effect would this have on their meal preparation plan?

Problem: Case: Annabelle Invest in the Market.

Annabelle Sizemore has cashed in some treasury bonds and a life insurance policy that her parents had accumulated over the years for her. She has also saved some money in certificates of deposit and savings bonds during the 10 years since she graduated from college. As a result, she has $120,000 available to invest. Given the recent rise in the stock market she feels that she should invest all of this amount there. She has researched the market and has decided that she wants to invest in an index fund tied to S&P stocks and in an Internet stock fund. However, she is very concerned about the volatility of Internet stocks. Therefore, she wants to balance her risk to some degree.

She has decided to select an index fund from Shield Securities, and an Internet stock fund from the Madison Funds, Inc. She has also decided that the proportion of the dollar amount she invests in the index fund relative to the Internet fend should be at least one-third, but that she should not invest more than twice the amount in the Internet fund that she invests in the index fund. The price per share of the index fund is $175, whereas the price per share for the Internet fund is $208. The average annual return during the last 3 years for the index fund has been \(17 \%\) and for the Internet stock fund it has been \(28 \% .\) She anticipates that both mutual funds will realize the same average returns for the coming year that they have in the recent past; however, at the end of the year she is likely to reevaluate her investment strategy anyway. Thus, she wants to develop an investment strategy that will maximize her return for the coming year.

Formulate a linear programming model for Annabelle that will indicate how much money she should invest in each fund and solve this model using the graphical method.

Suppose Annabelle decides to change her risk balancing formula by eliminating the restriction that the proportion of the amount she invests in the index fund to the amount that she invests in the Internet fund must be at least one-third. What will the effect be on her solution? Suppose instead that she eliminates the restriction that the proportion of money she invests in the Internet fund relative to the stock fund not exceed a ratio of 2 to 1 . How will this affect her solution?

If Annabelle can get one more dollar to invest, how will that affect her solution? Two more dollars? Three more dollars? What can you say about her return on her investment strategy given these successive changes?

Problem: Case: Mossaic Tiles, LTD

Gilbert Moss and Angela Pasaic spent several summers during their college years working at archaeological sites in the Southwest. While at these digs, they learned how to make ceramic tiles from local artisans. After college they made use of their college experiences to start a tile manufacturing firm called Mossaic Tiles, Ltd. They opened their plant in New Mexico, where they would have convenient access to a special clay they intend to use to make a clay derivative for their tiles. Their manufacturing operation consists of a few relatively simple but precarious steps, including molding the tiles, baking, and glazing.

Gilbert and Angela plan to produce two basic types of tile for use in home bathrooms, kitchens, sunrooms, and laundry rooms. The two types of tile are a larger, single-colored tile and a smaller, patterned tile. In the manufacturing process the color or pattern is added before a tile is glazed. Either a single color is sprayed over the top of a baked set of tiles or a stenciled pattern is sprayed on the top of a baked set of tiles.

The tiles are produced in batches of 100 . The first step is to pour the clay derivative into specially constructed molds. It takes 18 minutes to mold a batch of 100 larger tiles and 15 minutes to prepare a mold for a batch of 100 smaller tiles. The company has 60 hours available each week for molding. After the tiles are molded they are baked in a kiln: $0.27$ hour for a batch of 100 larger tiles and $0.58$ hour for a batch of 100 smaller tiles. The company has 105 hours available each week for baking. After baking, the tiles are either colored or patterned and glazed. This process takes $0.16$ hour for a batch of 100 larger tiles and $0.20$ hour for a batch of 100 smaller tiles. Forty hours are available each week for the glazing process. Each batch of 100 large tiles requires $32.8$ pounds of the clay derivative to produce, whereas each batch of smaller tiles requires 20 pounds. The company has 6,000 pounds of the clay derivative available each week.

Mossaic Tiles earns a profit of $190 for each batch of 100 of the larger tiles and $240 for each batch of 100 smaller patterned tiles. Angela and Gilbert want to know how many batches of each type of tile to produce each week to maximize profit. In addition, they also have some questions about resource usage they would like answered.

1. Formulate a linear programming model for Mossaic Tiles, Ltd. and determine the mix of the tiles it should manufacture each week.
2. Transform the model into standard form.
3. Solve the linear programming model graphically.
4. Determine the resources left over and not used at the optimal solution point.
5. Determine the sensitivity ranges for the objective function coefficients and constraint quantity values using the graphical solution of the model.
6. For artistic reasons Gilbert and Angela like to produce the smaller, patterned tiles better. They also believe in the long run the smaller tiles will be a more successful product. What must the profit be for the smaller tiles in order for the company to produce only the smaller tiles?
7. Solve the linear programming model using the computer and verify the sensitivity ranges computed in (E).
8. Mossaic believes it may be able to reduce the time required for molding to 16 minutes for a batch of larger tiles and 12 minutes for a batch of the smaller tiles. How wil] this affect the solution?

1. The company that provides Mossaic with clay has indicated that it can deliver an additional 100 pounds each week. Should Mossaic agree to this offer?

j. Mossaic is considering adding capacity to one of its kilns to provide 20 additional glazing hours per week at a cost of $\$ 90,000$. Should it make the investment?

K. The kiln for glazing had to be shut down for 3 hours, reducing the available kiln hours from 40 to 37 . What effect will this have on the solution?

Problem: Case: "The Possibility" Restaurant - Continued

In "The Possibility" Restaurant case problem in chapter 6 Angela Fox and Zooey Caulfield opened a French restaurant called "The Possibility." Initially, Angela and Zooey could not offer a full, varied menu, so their chef, Pierre, prepared two fullcourse dinners with beef and fish each evening. In the case problem, Angela and Zooey wanted to develop a linear programming model to help them determine the number of beef and fish meals they should prepare each night. Solve Zooey and Angela's linear programming model using the computer.

1. Angela and Zooey are considering investing in some advertising to increase the maximum number of meals they serve. They estimate that if they spend $30 per day on a newspaper ad it will increase the maximum number of meals they serve per day from 60 to 70. Should they make the investment?
2. Zooey and Angela are also concerned about the reliability of some of their kitchen staff. They estimate that on some evenings they could have a staff reduction of as much as five hours. How would this affect their profit level?
3. The final question they would like to explore is raising the price of the fish dinner. Angela believes the price for a fish dinner is a little low and that it could be closer to the price of a beef dinner without affecting customer demand. However, Zooey has noted that Pierre has already made plans based on the number of dinners recommended by the linear programming solution. Angela has suggested a price increase that will increase profit for the fish dinner to $14. Would this be acceptable to Pierre, and how much additional profit would be realized?

Problem: Case Problem: Julia’s Food Booth.

Julia Robertson is a senior at Tech and she's investigating different ways to finance her final year at school. She is considering leasing a food booth outside the Tech stadium at home football games. Tech sells out every home game and Julia knows, from attending the games herself, that everyone eats a lot of food. She has to pay $1,000 per game for a booth and the booths are not very large. Vendors can sell either food or drinks on Tech property, but not both. Only the Tech athletic department concession stands can sell both inside the stadium. She thinks slices of cheese pizza, hot dogs, and barbecue sandwiches are the most popular food items among fans and so these are the items she would sell. Most food items are sold during the hour before the game starts and during half time; thus it will not be possible for Julia to prepare the food while she is selling it. She must prepare the food ahead of time and then store it in a warming oven. For $600 she can lease a warming oven for the six-game home season. The oven has 16 shelves and each shelf is 3 feet by 4 feet. She plans to fill the oven with the three food items before the game and then again before half time.

Julia has negotiated with a local pizza delivery company to deliver 14-inch cheese pizzas twice each game-two hours before the game and right after the opening kickoff. Each pizza will cost her $6 and will include 8 slices. She estimates it will cost her $0.45 for each hot dog and $0.90 for each barbecue sandwich if she makes the barbecue herself the night before. She measured a hot dog and found it takes up about \(16 \mathrm{in}^{2}\) of space, whereas a barbecue sandwich takes up about \(25 \mathrm{in}^{2}\). She plans to sell a slice of pizza and a hot dog for $1.50 apiece and a barbecue sandwich for $2.25. She has $1,500 in cash available to purchase and prepare the food items for the first home game; for the remaining five games she will purchase her ingredients with money she has made from the previous game.

Price: $49.99

Solution: The downloadable solution consists of 25 pages, 3883 words and 21 charts.
Deliverable: Word Document