unlikely that a trial and error rounded solution will meet room availability limits. This means One way to improve the chances for a successful execution of the integer model is om For example, if the hospital has only two dental surgeons on a given day, no more than two the feasible ranes for the variables by taking into account the availability of other Test rooms (of any type) can be assigned to that department on that day, Setting tighter bounds may be effective in securing an optimal integer solution. Short of meeting this requirement, the only *8-1. Formulate the Fairville tax problem, assuming that the town council is specifying an ad- 8-2. The NW Shopping Mall conducts special events to attract potential patrons. Among the events that seem to attract teenagers, the young middle-aged group, and senior citiers, the two most popular are band concerts and art shows. Their costs per presentation are $1500 and $3000, respectively. The total (strict) annual budget allocated to the two events 8 Goal Programming Chapter that there is no alternative to imposing the integer condition. remaining option is to devise a heuristic for the problem. PROBLEMS Section Assigned Problems 8-1 to 8-11 8-12 to 8-21 8-22 to 8-25 8.1 8.2.1 8.2.2 ditional goal, Gs, that requires gasoline tax to equal at least 10% of the total tax bina is $15,000. The mall manager estimates the attendance as follows: Number attending per presentation Teenagers Young/middle-aged group Seniors Event 100 400 200 0 0 250 Band concert Art show The manager has set minimum goals of 1000, 1200, and 800 for the attendance of teenagers, the young/middle-aged group, and seniors, respectively. Formulate the problem as a GP model. *8-3. The Ozark University admission office is processing freshman applications for the up. coming academic year. The applications fall into three categories: in-state, out-of-state. and international. The male-female ratios for in-state and out-of-state applicants are 1:1 and 3:2, respectively. For international students, the corresponding ratio is 8:1. The American College Test (ACT) score is an important factor in accepting new students The statistics gathered by the university indicate that the average ACT scores for in-state, out-of-state, and international students are 27,26, and 23, respectively. The committee on admissions has established the following desirable goals for the new freshman class: (a) The incoming class is at least 1200 freshmen. (b) The average ACT score for all incoming students is at least 25. (c) International students constitute at least 10% of the incoming class. (a) Solve the problem, and specify whether or not all the goals are satisfied. (b) If, in addition, the size of the incoming class can be treated as a flexible goal that is twice as important as the ACT goal, how would this change affect requirements? 8-14. In the Ozark University admission situation described in Problem 8.3, suppose that the limit on the size of the incoming freshmen class must be met, but the remaining require- ments can be treated as flexible goals. Further, assume that the ACT score goal is twice as important as any of the remaining goals. 8-21. Solve Problem 8-20 using the Chebyshev method proposed in Problem 8-11. 8-22. In Example 8.2-2. suppose that the budget goal is increased to $110,000. The exposure *8-23. Solve Problem 8-1 using the following priority ordering for the goals: Problems 327 the solution? 48-15. In the Circle K model of Problem 8-2, is it possible to satisfy all the nutritional 3-16. In Problem 8-5, determine the solution, and specify whether or not the daily production of wheels and seats can be balanced, 3-17. In Problem 8-6, suppose that the market demand goal is twice as important as that of balancing the two machines, and that no overtime is allowed. Solve the problem, and h determine if the goals are met. 20 *8.18. In Problem 8-7, suppose that production strives to meet the quota for the two products, using overtime if necessary. Find a solution to the problem, and specify the amount of 1 overtime, if any, needed to meet the production quota. and 8-19. In the Vista City Hospital of Problem 8-8, suppose that only the bed limits represent flexible goals and that all the goals have equal weights. Can all the goals be met? 1 of 8-20. The Malco Company has compiled the following table from the files of five of its employees to study the impact on income of three factors: age, education (expressed in ave number of college years completed), and experience (expressed in number of years in ist the business) Experience Annual income Education ($) (year) (year) 30 39 44 48 37 reach a solution. Age (year) ate 4 5 2 0 3 5 10 14 18 9 40,000 48,000 38,000 36,000 41,000 te: ed fit into ed to ved Use the GP formulation in Problem 8-10 to fit the data into the linear equation y=bot bixi + b2x2 + bzxz. 1 is to 2 as a goal remains unchanged at 45 million persons. Show how the preemptive method will G > G > G > G > G. em. the twice as se interactive AMPL to solve Problems 8-22 to 8-25. unlikely that a trial and error rounded solution will meet room availability limits. This means One way to improve the chances for a successful execution of the integer model is om For example, if the hospital has only two dental surgeons on a given day, no more than two the feasible ranes for the variables by taking into account the availability of other Test rooms (of any type) can be assigned to that department on that day, Setting tighter bounds may be effective in securing an optimal integer solution. Short of meeting this requirement, the only *8-1. Formulate the Fairville tax problem, assuming that the town council is specifying an ad- 8-2. The NW Shopping Mall conducts special events to attract potential patrons. Among the events that seem to attract teenagers, the young middle-aged group, and senior citiers, the two most popular are band concerts and art shows. Their costs per presentation are $1500 and $3000, respectively. The total (strict) annual budget allocated to the two events 8 Goal Programming Chapter that there is no alternative to imposing the integer condition. remaining option is to devise a heuristic for the problem. PROBLEMS Section Assigned Problems 8-1 to 8-11 8-12 to 8-21 8-22 to 8-25 8.1 8.2.1 8.2.2 ditional goal, Gs, that requires gasoline tax to equal at least 10% of the total tax bina is $15,000. The mall manager estimates the attendance as follows: Number attending per presentation Teenagers Young/middle-aged group Seniors Event 100 400 200 0 0 250 Band concert Art show The manager has set minimum goals of 1000, 1200, and 800 for the attendance of teenagers, the young/middle-aged group, and seniors, respectively. Formulate the problem as a GP model. *8-3. The Ozark University admission office is processing freshman applications for the up. coming academic year. The applications fall into three categories: in-state, out-of-state. and international. The male-female ratios for in-state and out-of-state applicants are 1:1 and 3:2, respectively. For international students, the corresponding ratio is 8:1. The American College Test (ACT) score is an important factor in accepting new students The statistics gathered by the university indicate that the average ACT scores for in-state, out-of-state, and international students are 27,26, and 23, respectively. The committee on admissions has established the following desirable goals for the new freshman class: (a) The incoming class is at least 1200 freshmen. (b) The average ACT score for all incoming students is at least 25. (c) International students constitute at least 10% of the incoming class. (a) Solve the problem, and specify whether or not all the goals are satisfied. (b) If, in addition, the size of the incoming class can be treated as a flexible goal that is twice as important as the ACT goal, how would this change affect requirements? 8-14. In the Ozark University admission situation described in Problem 8.3, suppose that the limit on the size of the incoming freshmen class must be met, but the remaining require- ments can be treated as flexible goals. Further, assume that the ACT score goal is twice as important as any of the remaining goals. 8-21. Solve Problem 8-20 using the Chebyshev method proposed in Problem 8-11. 8-22. In Example 8.2-2. suppose that the budget goal is increased to $110,000. The exposure *8-23. Solve Problem 8-1 using the following priority ordering for the goals: Problems 327 the solution? 48-15. In the Circle K model of Problem 8-2, is it possible to satisfy all the nutritional 3-16. In Problem 8-5, determine the solution, and specify whether or not the daily production of wheels and seats can be balanced, 3-17. In Problem 8-6, suppose that the market demand goal is twice as important as that of balancing the two machines, and that no overtime is allowed. Solve the problem, and h determine if the goals are met. 20 *8.18. In Problem 8-7, suppose that production strives to meet the quota for the two products, using overtime if necessary. Find a solution to the problem, and specify the amount of 1 overtime, if any, needed to meet the production quota. and 8-19. In the Vista City Hospital of Problem 8-8, suppose that only the bed limits represent flexible goals and that all the goals have equal weights. Can all the goals be met? 1 of 8-20. The Malco Company has compiled the following table from the files of five of its employees to study the impact on income of three factors: age, education (expressed in ave number of college years completed), and experience (expressed in number of years in ist the business) Experience Annual income Education ($) (year) (year) 30 39 44 48 37 reach a solution. Age (year) ate 4 5 2 0 3 5 10 14 18 9 40,000 48,000 38,000 36,000 41,000 te: ed fit into ed to ved Use the GP formulation in Problem 8-10 to fit the data into the linear equation y=bot bixi + b2x2 + bzxz. 1 is to 2 as a goal remains unchanged at 45 million persons. Show how the preemptive method will G > G > G > G > G. em. the twice as se interactive AMPL to solve Problems 8-22 to 8-25