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6 6 KC DH 0 6 0 4 6 Problem 1:26 points Oxford University maintains a powerful mainframe computer research use by its faculty, Ph.D. students, and research associates. During all working hours, an operator must be available to operate and maintain the computer, as well as to perform some programming services. The director of the computer facility oversees the operation. It is now the beginning of the fall semester, and the director is confronted with the problem of assigning different working hours to her operators. Because all the operators are currently enrolled in the university, they are available to work only a limited number of hours each day. There are three operators. They all have different wage rates because of differences in their experience with computers and in their programming ability. The following table shows their wage rates, along with the maximum number of hours that each operator can work on each weekday. Maximum hours of Availability Operators Wage Rate Mon. Tue. Wed. Thurs. Fri. $10/hour 0 3 $10.1/hour 6 0 HB $9.9/hour 4 8 0 4 Each operator is guaranteed a certain minimum number of hours per week that will maintain an adequate knowledge of the operation. This level is set arbitrarily at 8 hours per week. That is, each operator should be assigned at least 8 hours in total through the week. The computer facility is to be open for operation from 10am to 4pm Monday through Friday with exactly one operator on duty during these hours. Therefore, for each weekday, the total number of operator hours available should be exactly equal to 6 hours. On Saturdays and Sundays, the computer is to be operated by other stuff. It is assumed that operators can work their hours during any period of time in a given day. That is, for instance, the director decides to have KC work 3.2 hours on Monday and HB for 2.8 hours on Monday, scheduling them is not of interest because they can work anytime during 10am-4pm on any weekday given that their time is not exceeding their available time on Monday. Because of a tight budget, the director must minimize the total operating cost. She wishes to determine the number of hours she should assign to each operator on each day. a) (15 points) Mathematically formulate a linear programming model for this problem by defining your decision variables and the notation you use and write your objective functions and constraints explicitly using your decision variables. b) (6 points, 3 points each) Answer the following yeso questions. Each question is independent of the others. Explain your answers clearly (2-3 sentences are sufficient; at most 4 sentences). i. Suppose that another operator (DK) is also available to work some hours on each weekday and DK has no restriction on minimum total number of hours he should work (i.e., no 8 hour limit for him). Can the minimum total operating cost increase? Yes or No. Explain your reasoning. ii. Suppose that, in addition to having each operator work at least 8 hours, the director has a restriction such that she cannot assign more than 12 hours in total to an operator through the week. Can the minimum total operating cost increase? Yes or No. Explain your reasoning. c) (5 points) Now suppose that there are n operators indexed by i = 1,2,3,..., n; and there are m days indexed by j = 1,2, ..., m. For day j, the facility should be open for exactly a; hours. Operator i should work at least bi hours throughout the m-day period. Also, operator i cannot work more than Cij hours on day j. Finally, operator i charges di per hour. Formulate this extended version of the problem in a compact form. Make sure to clearly write the ranges of the summations in your formulation. 6 6 KC DH 0 6 0 4 6 Problem 1:26 points Oxford University maintains a powerful mainframe computer research use by its faculty, Ph.D. students, and research associates. During all working hours, an operator must be available to operate and maintain the computer, as well as to perform some programming services. The director of the computer facility oversees the operation. It is now the beginning of the fall semester, and the director is confronted with the problem of assigning different working hours to her operators. Because all the operators are currently enrolled in the university, they are available to work only a limited number of hours each day. There are three operators. They all have different wage rates because of differences in their experience with computers and in their programming ability. The following table shows their wage rates, along with the maximum number of hours that each operator can work on each weekday. Maximum hours of Availability Operators Wage Rate Mon. Tue. Wed. Thurs. Fri. $10/hour 0 3 $10.1/hour 6 0 HB $9.9/hour 4 8 0 4 Each operator is guaranteed a certain minimum number of hours per week that will maintain an adequate knowledge of the operation. This level is set arbitrarily at 8 hours per week. That is, each operator should be assigned at least 8 hours in total through the week. The computer facility is to be open for operation from 10am to 4pm Monday through Friday with exactly one operator on duty during these hours. Therefore, for each weekday, the total number of operator hours available should be exactly equal to 6 hours. On Saturdays and Sundays, the computer is to be operated by other stuff. It is assumed that operators can work their hours during any period of time in a given day. That is, for instance, the director decides to have KC work 3.2 hours on Monday and HB for 2.8 hours on Monday, scheduling them is not of interest because they can work anytime during 10am-4pm on any weekday given that their time is not exceeding their available time on Monday. Because of a tight budget, the director must minimize the total operating cost. She wishes to determine the number of hours she should assign to each operator on each day. a) (15 points) Mathematically formulate a linear programming model for this problem by defining your decision variables and the notation you use and write your objective functions and constraints explicitly using your decision variables. b) (6 points, 3 points each) Answer the following yeso questions. Each question is independent of the others. Explain your answers clearly (2-3 sentences are sufficient; at most 4 sentences). i. Suppose that another operator (DK) is also available to work some hours on each weekday and DK has no restriction on minimum total number of hours he should work (i.e., no 8 hour limit for him). Can the minimum total operating cost increase? Yes or No. Explain your reasoning. ii. Suppose that, in addition to having each operator work at least 8 hours, the director has a restriction such that she cannot assign more than 12 hours in total to an operator through the week. Can the minimum total operating cost increase? Yes or No. Explain your reasoning. c) (5 points) Now suppose that there are n operators indexed by i = 1,2,3,..., n; and there are m days indexed by j = 1,2, ..., m. For day j, the facility should be open for exactly a; hours. Operator i should work at least bi hours throughout the m-day period. Also, operator i cannot work more than Cij hours on day j. Finally, operator i charges di per hour. Formulate this extended version of the problem in a compact form. Make sure to clearly write the ranges of the summations in your formulation

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