Mossaic Tiles, Ltd. Gilbert Moss and Angela Pasaic spent several summers during their college years working at archaeological sites in the Southwest. While at those 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 tiles for use in home bathrooms, kitchens, sunrooms, and laundry rooms. The two types of tiles are a larger, singlecolored 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 iles and 0.58 hour for a batch of 100 smaller tiles. The company has 105 hours available each week or baking. After baking, the tiles are either colored or patterned and glaze. This process takes 0.16 our for a batch of 100 larger tiles and 0.20 hour for a batch of 100 smaller tiles. Forty hours are vailable each week for the glazing process. Each batch of 100 large tiles requires 32.8 pounds of e clay derivative to produce, whereas each batch of smaller tiles requires 20 pounds. The company as 6,000 pounds of the clay derivative available each week. Mossaic Tiles earns a profit of $190 r 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 have some questions about resource usage they would like answered. A. Formulate a linear programming model for Mossaic Tiles, Ltd., and determine the mix of tiles it should manufacture each week. B. Transform the model into standard form. C. Solve the linear programming model graphically. D. Determine the resources left over and not used at the optimal solution point. E. Determine the sensitivity ranges for the objective function coefficients and constraint quantity values by using the computer(excel). F. For artistic reasons, Gilbert and Angela prefer to produce the smaller, patterned tiles. They also believe that 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? G. 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 smaller tiles. How will this affect the solution? H. 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? I. 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? The Metropolitan Police Department was recently 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 was assigned by the chief as head of a task force to find a way to determine the 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 that 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 staffing 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 was 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 every 60 minutes. He said that the difference was due to different traffic flows. i) Develop a linear programming model for this problem and solve it by using the graphical method. An organization conducts marketing research to learn about consumer characteristics, attitudes, and preferences. Marketing research firms that specialize in providing such information often do the actual research for client organizations. Typical services offered by a marketing research firm include designing the study, conducting market surveys, analyzing the data collected, and providing summary reports and recommendations for the client. In the research design phase, targets or quotas may be established for the number and types of respondents to be surveyed. The marketing research firm's objective is to conduct the survey so as to meet the client's needs at a minimum cost. Market Survey, Inc. (MSI), specializes in evaluating consumer reaction to new products, services, and advertising campaigns. A client firm requested MSI's assistance in ascertaining consumer reaction to a recently marketed household product. During meetings with the client, MSI agreed to conduct door-to-door personal interviews to obtain responses from households with children and households without children. In addition, MSI agreed to conduct both day and evening interviews. Specifically, the client's contract called for MSI to conduct 1000 interviews under the following quota guidelines. 1. Interview at least 400 households with children. 2. Interview at least 400 households without children. 3. The total number of households interviewed during the evening must be at least as great as the number of households interviewed during the day. 4. At least 40% of the interviews for households with children must be conducted during the evening. 5. At least 60% of the interviews for households without children must be conducted during the evening. Because the interviews for households with children take additional interviewer time and because evening interviewers are paid more than daytime interviewers, the cost varies with the type of interview. Based on previous research studies, estimates of the interview costs are as follows: (a) Define the decision variables for this problem (b) Provide a linear programming formulation to determine the household, time-of-day interview plan that will satisfy the contract requirements at a minimum total interviewing cost? (c) Solve the above LP model using MS Excel Solver and use your report to answer the following questions. i) What is the household, time-of-day interview plan that will satisfy the contract requirements and at what minimum total interviewing cost? ii) Which constraints are binding? iii) What are the dual values for the constraints? Interpret each. iv) Interpret the surplus variable, with a value of 200 and 40 respectively. v) The marketing decides to increase the per unit cost of daytime interviews of households with children to $50. What will be the effect of this decision on the optimal solution and total cost? Justify. The trust officer at Databank, needs to determine how to invest GHe200,000 in a collection of bonds to maximize the annual return. The bonds and their characteristics are shown in Table 1. Table 1: Bonds and their characteristics The manager wants to invest at most 50% of the money in long-term issues and at least 40% in low-risk issues. At least 60% of the funds should go into tax-free investments and at least 40% of the total annual return should be tax free. a) Formulate a linear programming model for this problem. b) Solve the above LP model using MS Excel Solver and use your report to answer the following questions 1) What is the optimal solution for this problem? 2) What is the maximum total annual return? 3) How much was invested in short-term bonds? 4) What is the total annual return from tax-free bonds? 5) If the total investment could be increased by GHC1,000, how would the total annual return be affected? 6) What will be the effect on the optimal solution and total annual return if the annual return of Bond B falls to 6%. 7) Interpret the reduced cost value for Bond D. What should be the minimum return on this bond to make it prudent to be invested in