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90-722: Management Science I, Final Exam, Spring 2013Do not turn the page until you are told to do so. Write your answers below each question.Note how much each problem is worth and budget your time. Total points possible: 80 + 5Name: ______________________________Q1: _________ out of 12 possible pointsQ2: _________ out of 5 possible pointsQ3: _________ out of 15 possible pointsQ4: _________ out of 12 possible pointsQ5: _________ out of 12 possible pointsQ6: _________ out of 24 possible pointsQ7: _________ out of 5 possible “bonus” pointsTotal Possible Points: 80 + 51Question #1: (12 Points): In the first half of the course, we only had to press “Solve” once for any given optimization (apart from sensitivity analysis). By contrast, in all three chapters since the midterm there have been reasons to optimize multiple times for any given problem. In a few concise, complete, and legible sentences explainwhy that is so for:a) Integer programming and different levels of the “tolerance” parameter b) Goal programming and varying weights on deviational variablesc) Nonlinear programming with the GRG algorithm and various initial values for the decision variablesQuestion #2: (5 Points): Why does the GRG algorithm for NLPs dislike plateaus?2Question #3: (15 Points): Below are snapshots of the node portion (i.e., right hand side) of a node-arc incidence approach to modeling network optimization problems in a spreadsheet. Identify as precisely as possible what type of problem is being solved in each case (where by type of problem I mean something like “product mix”, “make vs. buy”, “allocation”, etc.). 3Question #4: (12 Points): The SS solution to the MasterDebt Lockbox problem is pasted below. Recall that the challenge was to decide which of the six lockboxes to operate, and which regions to assign to each lockbox, in order to minimize the combined cost of (1) lockbox operation plus (2) “float”. a) Where (in what cities) are lockboxes being operated in this solution?b) What is the cost of float in this solution? (Give the answer in thousands of dollars.)c) Part of the logic of the model is “If any region is assigned to a given lockbox, then we must operate that lockbox and pay the associated price,” but that logic is implemented with linear equations rather than with an =IF() statement to make things easier on Solver. What constraint would you add to Solver to implement this logic? (Specify cell range on left and right hand side and whether the comparator is ≥ , ≤ or, =.)d) Write the algebraic expression(s) which correspond to the constraint described in part c.(I.e., parts c & d ask about the same constraint. Part c asks you to state it in “Solver speak”; d asks you toexpress it algebraically.) 4Answer part a)Answer part b)Answer part c)Answer part d)5Question #5: (12 Points): Suppose Branch & Bound were working on a minimization problem and its tree currently has eight activenodes whose LP relaxations yield the following results (a “?” for Z*IP indicates the solution to the LP relaxation was non-integer.)a) Which subproblems can be pruned without jeopardizing the ability to find an optimal solution?b) Which additional subproblems could be pruned if the tolerance parameter were set to 20%?c) What are the tightest bounds you can provide on the original problem’s optimal solution value? 6Question #6: (24 Points): This problem pertains to assigning students to systems project groups. Use the notation that Xij indicates whether student i ends up in group j. Assume the number of students and groups is fixed throughout this problem, and do not worry about specifying their values. a) If parameter lij denotes how much student i would like (the letter “l” for like) being on project j (big numbers being good), formulate an LP that maximizes total student happiness subject to the constraint that every project has at least 7 and no more than 14 students.b) Suppose pij indicates how much the professor of course j would prefer (p for prefer) having student i in his or her project group. Suppose the Dean defines target levels for the happiness of students and professors as t1 and t2, respectively. Write soft (or goal) constraints for the twin goals of making students and professors as happy as possible.c) Suppose the Dean did not know how to pick target values. What would Ragsdale suggest based on the MOLP portions of Chapter 7?7d) Write a GP objective function for the Dean that penalizes the relevant deviation from the student target three times as much as the corresponding deviation from the faculty target – in both cases thinking of deviations in percentage terms.e) Suppose the Dean did not want to specify the relative importance of the two goals, but instead wanted to see the full range of possibilities produced by varying the relative weight placed on them. Sketch the resulting figure you would produce for the Dean, paying attention to the slope of the efficient frontier. (Don’t sweat the curvature, meaning whether the frontier is convex or concave. I can tell you it’ll be concave.) f) The Dean has a third goal of holding down the maximum size of systems classes. So far in this problem that upper limit has been 14. So label the efficient frontier from part e with “max size = 14”. Now draw a dashed line on the same graph as in part e for the frontier with “max size = 12”,indicating which side of the original frontier it would be on. (Make the original frontier be a solidline so it is easy to distinguish from this new, dashed line.)8Question #7: (5 Points “Bonus” – do last): Referring to the situation in the previous problem, suppose the Dean wanted to minimize the maximum size of a systems class subject to the (hard) constraints that student and professor satisfaction each be at least 75% of their target values of t1 and t2, respectively. Formulate this problem for the Dean as an

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