Introduction to Management Science Study Set 3

Business

Quiz 9 :

Multicriteria Decision Making

Quiz 9 :

Multicriteria Decision Making

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The Oregon Atlantic Company produces two kinds of paper-newsprint and white wrapping paper (butcher paper). It requires 5 minutes to produce a yard of newsprint and 8 minutes to produce a yard of wrapping paper. The company has 4,800 minutes of normal production capacity available each week. The profit is $0.20 for a yard of newsprint and $0.25 for a yard of wrapping paper. The weekly demand is for 500 yards of newsprint and 400 yards of wrapping paper. The company has established the following goals, in order of priority: (1) Limit overtime to 480 minutes. (2) Achieve a profit of $300 each week. (3) Fulfill the demand for the products in order of magnitude of their profits. (4) Avoid underutilization of production capacity. a. Formulate a goal programming model to determine the number of yards of each type of paper to produce weekly to satisfy the various goals. b. Solve the goal programming model by using the computer.
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In Problem in Chapter, Computers Unlimited sells microcomputers and distributes them from three warehouses to four universities. The available supply at the three warehouses, demand at the four universities, and shipping costs are shown in the following table: img Instead of its original objective of cost minimization, Computers Unlimited has indicated the following goals, arranged in order of their importance: (1) A M has been one of its better long-term customers, so Computers Unlimited wants to meet all of A M 's demands. (2) Because of recent problems with a trucking union, it wants to ship at least 80 units from the Washington warehouse to Central University. (3) To maintain the best possible relations with all its customers, Computers Unlimited would like to meet no less than 80% of each customer's demand. (4) It would like to keep total transportation costs to no more than 110% of the $22,470 total cost achieved with the optimal allocation, using the transportation solution method. (5) Because of dissatisfaction with the trucking firm it uses for the Atlanta-to-State deliveries, it would like to minimize the number of units shipped over this route. a. Formulate a goal programming model for this problem to determine the number of microcomputers to ship on each route to achieve the goals. b. Solve this model by using the computer. Problem Computers Unlimited sells microcomputers to universities and colleges on the East Coast and ships them from three distribution warehouses. The firm is able to supply the following numbers of microcomputers to the universities by the beginning of the academic year: img Four universities have ordered microcomputers that must be delivered and installed by the beginning of the academic year: img The shipping and installation costs per microcomputer from each distributor to each university are as follows: img Solve this problem by using the computer.
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First, put together the linear programming model; where the destinations are represented with A, B, C and D. The sources are 1-3.
Maximize
img This would be subject to the following equations
img Representing the supply
img Representing the demand
Where,
img Now, to solve this transportation problem, use the QM for Window software (provided with the book) following these instructions:
First, open the software; under module, select Transportation. Next, open a blank document. A menu will open; enter the document title if needed. Then, enter the number of Sources (1, 2 and 3) which is 3 for this problem. Enter the number of destinations which is 4 for this problem ( A, B, C and D ). Select OK and a window will show in order to enter the equations. Enter the values for Destination and Sources; second enter the given supply and demand numbers (3 and 4 for this problem); then, select solve on the top right corner and a series of windows with solutions. The window titled: Transportation Shipments has the values of the variables.
From source 1 to destination B the value is 250
From source 1 to destination D the value is 170
From source 2 to destination A the value is 520
From source 2 to destination C the value is 90
From source 3 to destination C the value is 130
From source 3 to destination D the value is 210
Hence,
img img

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The Wearever Carpet Company manufactures two brands of carpet-shag and sculptured-in 100-yard lots. It requires 8 hours to produce one lot of shag carpet and 6 hours to produce one lot of sculptured carpet. The company has the following production goals, in prioritized order: (1) Do not underutilize production capacity, which is 480 hours. (2) Achieve product demand of 40 (100-yard) lots for shag and 50 (100-yard) lots for sculptured carpet. Meeting demand for shag is more important than meeting demand for sculptured, by a ratio of 5 to 2. (3) Limit production overtime to 20 hours. a. Formulate a goal programming model to determine the amount of shag and sculptured carpet to produce to best meet the company's goals. b. Solve this model by using the computer.
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First, the integer programming model needs to be formulated and then solved. Integer programming model should be written using the slack variables in order to have = signs in order to exchange for the
img signs (this is at most, hence use
img ).
Minimize (Units should be used)
img Then consider the constrains,
img Now, solve the equations using the QM for Window software following these instructions:
First, open the software; under module, select goal programming. Next, open a blank document. A menu will open; enter the document title if needed. Then, enter the number of Constrains which is 3 for this problem. Enter the number of variables which is 3 for this problem. Select OK and a window will show in order to enter the equations. The three bottom equations are the Constraints and the coefficients for the top equations go under the Maximize value. Also, enter the corresponding sing (
img ; then, select solve on the top right corner and a series of windows with solutions will open.
The following shows the solutions in the Final Tableau window, where
img represents labor underutilization and
img represents overtime:
img The equations with added variables are as follows:
img

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Mac's Warehouse is a large discount store that operates 7 days per week. The store needs the following number of full-time employees working each day of the week: img Each employee must work 5 consecutive days each week and then have 2 days off. For example, any employee who works Sunday through Thursday has Friday and Saturday off. The store currently has a total of 60 employees available to work. Mac's has developed the following set of prioritized goals for employee scheduling: (1) The store would like to avoid hiring any additional employees. (2) The most important days for the store to be fully staffed are Saturday and Sunday. (3) The next most important day to be fully staffed is Friday. (4) The store would like to be fully staffed the remaining 4 days in the week. a. Formulate a goal programming model to determine the number of employees who should begin their 5-day workweek each day of the week to achieve the store's objectives. b. Solve this model by using the computer.
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A rural clinic hires its staff from nearby cities and towns on a part-time basis. The clinic attempts to have a general practitioner (GP), a nurse, and an internist on duty during at least a portion of each week. The clinic has a weekly budget of $1,200. A GP charges the clinic $40 per hour, a nurse charges $20 per hour, and an internist charges $150 per hour. The clinic has established the following goals, in order of priority: (1) A nurse should be available at least 30 hours per week. (2) The weekly budget of $1,200 should not be exceeded. (3) A GP or an internist should be available at least 20 hours per week. (4) An internist should be available at least 6 hours per week. a. Formulate a goal programming model to determine the number of hours to hire each staff member to satisfy the various goals. b. Solve the model by using the computer.
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A farmer in the Midwest has 1,000 acres of land on which she intends to plant corn, wheat, and soybeans. Each acre of corn costs $100 for preparation, requires 7 worker-days of labor, and yields a profit of $30. An acre of wheat costs $120 to prepare, requires 10 worker-days of labor, and yields $40 profit. An acre of soybeans costs $70 to prepare, requires 8 worker-days, and yields $20 profit. The farmer has taken out a loan of $80,000 for crop preparation and has contracted with a union for 6,000 worker-days of labor. A midwestern granary has agreed to purchase 200 acres of corn, 500 acres of wheat, and 300 acres of soybeans. The farmer has established the following goals, in order of their importance: (1) To maintain good relations with the union, the labor contract must be honored; that is, the full 6,000 worker-days of labor contracted for must be used. (2) Preparation costs should not exceed the loan amount so that additional loans will not have to be secured. (3) The farmer desires a profit of at least $105,000 to remain in good financial condition. (4) Contracting for excess labor should be avoided. (5) The farmer would like to use as much of the available acreage as possible. (6) The farmer would like to meet the sales agreement with the granary. However, the goal should be weighted according to the profit returned by each crop. a. Formulate a goal programming model to determine the number of acres of each crop the farmer should plant to satisfy the goals in the best possible way. b. Solve this model by using the computer.
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Broderick Crawford is the district commander for the Catawba Valley highway patrol district in western Pennsylvania. He is attempting to assign highway patrol cars to different road segments in his district. The primary function of the highway patrol force is to patrol roads outside incorporated city and town limits in the district to deter traffic violators and accidents. This objective is typically achieved by maintaining a visible presence- letting motorists see patrol units on a regular basis and giving out warnings, citations, and so forth. Secondary activities of a patrol unit include providing assistance to motorists, answering distress calls, handling emergencies and accidents when called to the scene, and occasionally apprehending criminals. Commander Crawford has 23 patrol cars that he wants to assign to the following six major road segment areas: road segment 1, interstate, north road segment 2, urban area, north road segment 3, four-lane highway, east road segment 4, two-lane highway, west road segment 5, interstate/four-lane highway, south road segment 6, two-lane highway (heavy truck traffic), south Each of these road segments includes the primary arteries, as indicated earlier, plus adjoining roads. All the road segments have different levels of traffic density and accident rates, which are key factors in determining how many patrol units to assign. However, these factors do not always coincide. For example, interstate highway segments typically have high traffic density but low accident rates, whereas some two-lane highways have low traffic density but high accident rates. Differences often occur because of variations in road conditions (such as sharp curves, visibility, and width). Other conditions, such as heavy truck traffic (as on segment 6), also contribute to high accident rates. Each segment requires different operating costs, including maintenance and repair, fuel, and so on because of different operating conditions. The commander's most pressing objective is to limit daily operating costs to $450. The daily operating costs per road segment are as follows: img The commander would like to reduce the accident rate for the district as well as increase both physical and sight contacts, which are deterrents to potential traffic violators. The commander would also like to achieve a reasonable average response time for a patrol unit to respond to a call for each road segment. The average accident rate reduction (per million miles traveled) and physical contacts and sight contacts per car for each road segment are shown in the following table: img The commander's second-most-important goal is to reduce the average accident rate for the district by five accidents per million miles traveled. The commander's next goals (in order) are to achieve 350 physical contacts and 30,000 sight contacts per day in the district. If no patrol units are assigned in the district, the average time to respond to a distress call anywhere in the district from the main district headquarters and motor pool is 28 minutes. Each car assigned to a road segment reduces the overall average response time in the district by the following amounts: img The commander's last objective is to achieve an average response time to distress calls of 15 minutes. Because of local and political pressure, the commander has to assign at least two patrol units to each road segment. In addition, the commander believes that a maximum of five patrol units is sufficient for any particular road segment. Formulate and solve a goal programming model to determine the number of patrol units to assign to each road segment to achieve the commander's goals.
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Solve the following goal programming model by using the computer: img
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The Bay City Parks and Recreation Department has received a federal grant of $600,000 to expand its public recreation facilities. City council representatives have demanded four different types of facilities-gymnasiums, athletic fields, tennis courts, and swimming pools. In fact, the demand by various communities in the city has been for 7 gyms, 10 athletic fields, 8 tennis courts, and 12 swimming pools. Each facility costs a certain amount, requires a certain number of acres, and is expected to be used a certain amount, as follows: img The Parks and Recreation Department has located 50 acres of land for construction (although more land could be located, if necessary). The department has established the following goals, listed in order of their priority: (1) The department wants to spend the total grant because any amount not spent must be returned to the government. (2) The department wants the facilities to be used by a total of at least 20,000 people each week. (3) The department wants to avoid having to secure more than the 50 acres of land already located. (4) The department would like to meet the demands of the city council for new facilities. However, this goal should be weighted according to the number of people expected to use each facility. a. Formulate a goal programming model to determine how many of each type of facility should be constructed to best achieve the city's goals. b. Solve this model by using the computer so that the solution values are integers.
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The Barrett Textile Mill was checked by inspectors enforcing Occupational Safety and Health Administration (OSHA) codes. The inspectors found violations in four categories: hazardous materials, fire protection, hand-powered tools, and machine guarding. In each category the mill was not in 100% compliance. Each percentage point of increase in the compliance level in each category will reduce the frequency of accidents, decrease the accident cost per worker, and constitute progress toward satisfying the OSHA compliance level. However, achieving compliance does cost the mill money. The following table shows the benefits (in accident frequency and accident cost per worker) and the costs of a percentage point increase in compliance in each category: img To achieve 100% compliance in all four categories, the mill will have to increase compliance in hazardous materials by 60 percentage points (i.e., it is now at 40% compliance), in fire protection by 28 percentage points, in hand-powered tools by 35 percentage points, and in machine guarding by 17 percentage points. However, the management of the mill faces a dilemma, in that only $52,000 is available to spend on safety. Any larger expenditure could jeopardize the financial standing of the mill. Thus, management hopes to achieve a level of accident reduction and compliance that is within the company's budget limitation and that will satisfy OSHA authorities enough to temporarily delay punitive action. Therefore, management has established four goals, listed here in order of importance: (1) Do not exceed the budget constraint of $52,000. (2) Achieve the percentage increases in compliance necessary to achieve 100% compliance in each category. (3) Achieve total accident frequency reduction of 20 accidents/105 hours of exposure. (This goal denotes management's desire to minimize the frequency of accidents even if 100% compliance cannot be achieved in all categories.) (4) Reduce the total accident cost per worker by $115. a. Formulate a goal programming model to determine the percentage points of compliance needed in each category to satisfy the goals. b. Solve this model by using the computer.
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Katherine Miller is a senior in the department of information technology at Tech. For the past few months she has been involved in the job search process. She has an excellent résumé, with a high grade point average and a strong record of campus participation in clubs and activities. As a result, she has had a number of good interviews with various companies. She now has job offers from five companies-American Systems Developers, Anderssun Consulting, National Computing Software Systems (NCSS), the Gulf-South Company, and Electronic Village. American Systems Developers and Anderssun Consulting are both large national consulting firms with offices in several major cities. If Katherine accepted the offer of either of these firms, she would primarily work on project teams assigned to develop decision support and information systems for corporate clients around the country. If she went with American Systems Developers, her home base would be in Atlanta, and if she accepted Anderssun's offer, she would be located in Washington, DC. However, in both cases she would be traveling a great deal and could sometimes be on the road at a client location for as much as 6 to 9 months. NCSS is a software and computer systems development company with a campus-like location in Chicago. Although her job with NCSS would involve some traveling, it would never be more than several weeks at any one time. Gulf-South is a bank holding company that operates eight different banks and its various branches in six southeastern states. If Katherine accepted Gulf-South's offer, she would be located in Tampa, where she would work in operations systems. She would be involved in developing information and support systems for bank operations, and she would have minimal travel. Electronic Village is a national chain of discount stores specializing in electronic products, such as televisions, DVRs, CD players, MP3 players, and computers. Her job with Electronic Village would be at its corporate headquarters in Nashville, Tennessee, where she would develop and maintain computer systems to be used for inventory control at the hundreds of Electronic Village stores across the country. She would be required to travel very little. American Systems Developers has offered Katherine a starting salary of $38,000 annually, and Anderssun Consulting has offered her $41,000 per year. NCSS has offered her an annual salary of $46,000, whereas Gulf- South has offered her $35,000 per year, and Electronic Village has offered her a salary of $32,000 per year. Katherine is having a difficult time making her decision. All the companies have excellent reputations, are financially healthy, and have good opportunities for advancement. All are demanding in terms of the workload they require. All five companies have given Katherine only a few weeks to make a decision regarding their offers. Katherine has decided to use the analytical hierarchy process (AHP) to help decide which job offer she should accept. She has developed a list of criteria that are important to her in deciding which job to take. The criteria, in no particular order, are (1) salary; (2) cost of living in the city where she would be located; (3) amount of travel associated with her job; (4) climate (weather) where she would be located; (5) entertainment and cultural opportunities, including sports, theater, museums, parks, and so on; (6) universities where she can work on an MBA degree part time and at night; (7) the crime rate in the city where she would live; (8) the nature of the job and what she would be doing; and (9) her proximity to friends and relatives. Katherine realizes that she has very limited information on which to compare the five jobs, based on most of these criteria, so she knows she needs to go to the library and do some research, especially on the five different job locations. Put yourself in Katherine's shoes and, using all or some of her criteria and your own preferences and knowledge, develop an overall ranking of the jobs, using AHP.
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The East Midvale Textile Company produces denim and brushed-cotton cloth. The average production rate for both types of cloth is 1,000 yards per hour, and the normal weekly production capacity (running two shifts) is 80 hours. The marketing department estimates that the maximum weekly demand is for 60,000 yards of denim and 35,000 yards of brushed cotton. The profit is $3.00 per yard for denim and $2.00 per yard for brushed cotton. The company has established the following four goals, listed in order of importance: (1) Eliminate underutilization of production capacity to maintain stable employment levels. (2) Limit overtime to 10 hours. (3) Meet demand for denim and brushed cotton weighted according to profit for each. (4) Minimize overtime as much as possible. a. Formulate a goal programming model to determine the number of yards (in 1,000-yard lots) to produce to satisfy the goals. b. Solve this model by using the computer.
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The Eaststate Manufacturing Company produces four different airplane parts from fabricated sheet metal for several major aircraft companies. The manufacturing process consists of four operations-stamping, assembly, finishing, and packaging. The processing times per unit for each operation and total available hours per year to produce these parts are as follows: img The sheet metal required for each part, the estimated annual demand, and the profit per part are as follows: img The company has 15,000 square feet of fabricated metal delivered each month. The company has the following prioritized production goals: (1) Avoid overtime, which would erode profit levels. (2) Meet parts demand. (3) Achieve an annual profit of $700,000. (4) Avoid ordering more material because a surcharge is required by the supplier for changing the standard monthly order. a. Formulate a goal programming model to determine the amount of each part to produce to achieve the company's objectives. b. Solve this model by using the computer. c. How would the solution be affected if the first two priorities were reversed
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The Growall Fertilizer Company produces three types of fertilizer-Supergro, Dynaplant, and Soilsaver. The company has the capacity to produce a maximum of 2,000 tons of fertilizer in a week. It costs $800 to produce a ton of Supergro, $1,500 for Dynaplant, and $500 for Soilsaver. The production process requires 10 hours of labor for a ton of Supergro, 12 hours for a ton of Dynaplant, and 18 hours for a ton of Soilsaver. The company has 800 hours of normal production labor available each week. Each week the company can expect a demand for 800 tons of Supergro, 900 tons of Dynaplant, and 1,100 tons of Soilsaver. The company has established the following goals, in order of their priority: (1) The company does not want to spend over $20,000 per week on production, if possible. (2) The company would like to limit overtime to 100 hours per week. (3) The company wants to meet demand for all three fertilizers; however, it is twice as important to meet the demand for Supergro as it is to meet the demand for Dynaplant, and it is twice as important to meet the demand for Dynaplant as it is to meet the demand for Soilsaver. (4) It is desirable to avoid producing under capacity, if possible. (5) Because of union agreements, the company wants to avoid underutilization of labor. a. Formulate a goal programming model to determine the number of tons of each brand of fertilizer to produce to satisfy the goals. b. Solve this model by using the computer.
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A manufacturing company produces products 1, 2, and 3. The three products have the following resource requirements and produce the following profit: img At present, the firm has a daily labor capacity of 240 available hours and a daily supply of 400 pounds of material. The general linear programming formulation for this problem is as follows: img Management has developed the following set of goals, arranged in order of their importance to the firm: (1) Because of recent labor relations difficulties, management wants to avoid underutilization of normal production capacity. (2) Management has established a satisfactory profit level of $500 per day. (3) Overtime is to be minimized as much as possible. (4) Management wants to minimize the purchase of additional materials to avoid handling and storage problems. Formulate a goal programming model to determine the number of each product to produce to best satisfy the goals.
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Each November the Baseball Writers' Association of America (BBWAA) releases its list of that year's nominees for induction into the National Baseball Hall of Fame in Cooperstown, New York. There are 575 voting members of the BBWAA, and candidates must be named on 75% of the ballots cast to gain entry into the National Baseball Hall of Fame. There is no limit to the number of players who can be nominated or who can be elected. Go to the BBWAA Web site at http://bbwaa.com/ and from the list of this year's nominees select the five or six "field players" (as opposed to pitchers) who you think are the top candidates for selection, and develop an AHP model using your own criteria to rank these candidates. Criteria might include, among other things, hits, lifetime batting average, home runs, runs batted in, runs, stolen bases, awards, years played, fielding percentage, number of World Series participated in, and a comparison with Hall of Famers overall and at their field position (e.g., catcher, first base, outfield). Player bios including their playing statistics can be obtained by searching the Internet.
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Solve the following goal programming model graphically and by using the computer: img
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Solve the following goal programming model graphically and by using the computer: img
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The Oakdale County School Board was meeting in special session. A federal judge had ordered the board to present an acceptable busing plan for racially balancing the four high schools in Oakdale County within a week. The judge had previously given the school board several opportunities to informally present a plan, but the members had been unable to agree among themselves. Every time they met and started to develop a plan to bus students from one high school district to another, an argument would arise before they got past the first busing move, and they would adjourn the meeting. This time, however, they knew the judge had lost patience and they had to agree on something. Of the four schools, only West High School was racially balanced, with 500 white students and 500 black students. North High School had 1,000 white students and only 300 black students; East High School was almost as bad, with 1,050 white students and 400 black students. South High School was predominantly black, with 800 black students and 450 white students. Overall, of the 5,000 students in Oakdale County, 60% were white and 40% black. "Look," said John Connor, a school board member from the West district, "rather than starting off by trying to shift students from one district to another, why don't we try to establish what we want to accomplish-you know, what our goals are" Several of the other members nodded in agreement, and Fred Harvey, the board chairman, said, "Good idea, John." "Okay, the first goal seems pretty evident to me," John said. "Sixty percent of our students are white, and 40% are black, so that's what we need our schools to be, 60% and 40%." "That's okay for you to say, John," Betty Philips argued, "because your district has those proportions already-so you won't have any busing. But my district in the North is a long way from that ratio, and we would have to bus a lot of our students to achieve a 60%/40% ratio." " I'm not saying it, Betty," said John. "That is basically what Judge Barry has been saying for 6 months." "John's right, Betty, and we're not busing students yet; we're just putting down our objectives," said Fred. "I think that has to be our highest-priority objective. How about the rest of you" They all nodded their agreement, even Betty Philips, reluctantly. "Since we know we're going to have to bus students to achieve this ratio at each school, I think we ought to try to minimize the amount of traveling the students will have to do," suggested Mickey Gibboney, a member from the South district. Fred Harvey noted that page 10 of their handout had a chart showing the average mileage a student in one district would have to travel on a bus to the high school in each of the other districts. The chart looked like this: img "Why don't we try to set some reasonable objectives for total busing miles, for the students' sake and for budgeting reasons" Cassandra Watkins asked. "I would suggest about 30,000 miles per day, based on the miles we bus students now. If we get much higher than that, we're not going to have the money to pay for it, and it means we'll be busing students all over the place." The other members nodded and agreed. "Okay," said Fred Harvey, "that'll be our number-two goal." Betty Philips spoke up again. "I'll tell you another thing I don't want to see happen, and that's any more overcrowding at North High School. We have 100 students more than capacity now." "You think you have problems!" Bob Wilson exclaimed. "In East we have 1,450 students and capacity for 1,000. I think no overcrowding is a great idea!" "I agree," said Mickey Gibboney. "We're 250 over our capacity at South High School." "That's a nice idea," John Connor responded, "and I realize that we have 200 students less than our capacity at West High School. However, let's face it, in the county we have capacity for 4,400, not 5,000, students, so there's going to be some overcrowding. I think our objective should be that all four schools share in the overcrowding proportionally." "That sounds reasonable to me," said Fred Harvey. "How about the rest of you Okay to say our number-three goal is to be as close to capacity at each school as possible but share proportionally in the overcrowding" They all voiced their approval. "Well," John Connor concluded, "I think we have identified the things we want to accomplish in our plan. Now if we could just use some magic trick to find a plan for busing students between the districts that would achieve all these goals." The others nodded and frowned. A. Formulate a goal programming model to help the board with its dilemma. B. Solve the goal programming model by using the computer.
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Suntrek is a global denim fabric and jeans manufacturer. It purchases raw, baled cotton from locations around the world (primarily the United States and Mexico) and ships it (in containers) to various factories it operates to manufacture denim fabric and jeans, primarily in the Asia-Pacific and Indian Ocean areas. It also ships some of the fabric to other factories where it produces denim jeans. It ultimately ships all of the jeans (and a small amount of fabric) to retail clothing companies primarily in the United States, Europe, and Japan. It is considering building a new denim jeans factory with a warehousing and distribution center for baled cotton, denim fabric, and denim jeans. The five possible port locations for the new facility are Da Nang, Mumbai, Manila, Shanghai, and Karachi. Suntrek's weighted decision criteria are labor wage rates (0.20), cotton and fabric supply (0.18), shipping/transport availability and cost (0.15), infrastructure (0.12), risk (0.10), import/export regulations and tariffs (0.09), government stability (0.06), labor sustainability (0.05), and environmental sustainability (0.05). Using the Internet, research the five possible port locations and assign a grade for each location for each criterion, and determine an overall ranking of the locations for Suntrek.
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