90-722: Management Science I, Final Exam (Afternoon), Spring 2019Do 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: 100Name: _________________________________ Section: __________Q1: _________ out of 12 possible pointsQ2: _________ out of 18 possible pointsQ3: _________ out of 12 possible pointsQ4: _________ out of 20 possible pointsQ5: _________ out of 17 possible pointsQ6: _________ out of 11 possible pointsQ7: _________ out of 10 possible pointsTotal Possible Points: 1001Question #1: (12 points). Provide short, clear, and neatly written answers to these questions. Correctideas poorly communicated will not receive credit. Think before you write. a) What is the role of the 2nd objective when addressing multiple objectives via lexicographicpreferences?b) If you use a dummy arc to convert a max flow problem to a transshipment problem, what shouldyou tell Excel is the demand at the nodes at either end of that arc?c) How many decision variables and constraints (besides non-negativity) are there in a standardmake vs. buy problem with 4 products and 3 resource constraints? d) What is the sign of the shadow price on a binding greater than or equal to constraint for an LPminimization problem? e) In the bottom half of Solver’s LP sensitivity report, what is it that can change within the rangespecified by the allowable increase and decrease? That is, what variable, parameter, or conceptdoes that allowable range apply to?f) What is nonlinear in the Chapter 8 portfolio optimization model? 2Question #2: (18 Points):What algorithm or method that we discussed in class would be best for solving a:Capital budgeting problem: ___________________________________________________Traveling Salesman Problem (TSP): ____________________________________________Equipment Replacement Problem: ______________________________________________Make vs. Buy problems as discussed in Chapter 3: ___________________________________Convex nonlinear optimization problem: __________________________________________General Nonlinear optimization with two decision variables: __________________________3Question #3: (12 Points): Consider the branch and bound tree depicted below.a) Is the problem a maximization or minimization problem?b) Which subproblem(s) cannot contain the optimal solution?c) Which additional subproblem(s) could be bounded if the tolerance parameter were set at 10%?d) What are the tightest bounds you can provide on the optimal solution value to the original problem?4Question #4: (20 points): Suppose you need to acquire 6,000 units of something (e.g., hot tubs) from amixture of ten suppliers (e.g., “factories”). Each source has its own fixed and variable cost. E.g., if youordered 100 units from Supplier #1 the cost would be $900 + 28 * $100. Suppliers have limitedcapacities and none will accept an order of fewer than 100 units.Formulate an integer linear program to minimize the cost of obtaining the 6,000 items subject to twoconstraints:#1: At least 2,000 are obtained from preferred suppliers#2: At least seven sources are used (in order to avoid being too dependent on a few suppliers)You are strongly encouraged to use summation notation and to use the following parameter definitions(which you do not need to rewrite)FCi = fixed cost of ordering any from supplier i,VCi = variable cost per item ordered from supplier i, andCi = maximum number that can be ordered from supplier i,5Question #5: (17 points) Explain how you would extend that algebraic formulation if you have three goals:#1: Minimize the cost of obtaining the 6,000 units. #2: Obtain 3,150 units from the first three suppliers who are preferred.#3: Source from all ten suppliers to diversify sources and associated risks. In particular, write the algebraic expressions for what you would add to create a linear integer goalprogram that minimizes the maximum undesirable percent deviation from these targets, using softconstraints with both undershoot and overshoot deviational variables. You may use summation notationto sum over suppliers but not for the goals. Use the ti notation only for goals for which no numericaltarget is specified; if a numerical target is specified, write that number. 6Question #6: (11 Points): The midterm asked questions based on an augmented version of Howie’s Hot Tub problem in which he can make two additional models, the deluxe Vortices and the cut rate Sink Holes. Its algebraic formulation was:Max Z = 350 X1 + 300 X2 + 625 X3 + 99 X4s.t. X1 + X2 + 2 X3 + X4 ≤ 2009 X1 + 6 X2 + 12 X3 + X4 ≤ 156612 X1 + 16 X2 + 40 X3 + X4 ≤ 2880Xi ≥ 0 for i=1,2,3,4I solved an extended version of that problem in which: (a) Howie must make at least two types ofHot Tubs, (b) if he makes any of a model he must make at least 50 of that model, (c) there are fixed costs of $15,000 if he produces any Aqua Spas, $12,000 if he produces any Hydroluxes, $9,000 if he produces any Vortices, and $1,000 if he produces any Sink Holes. The screen shot isshown below, after solving to optimality.7Show how the Solver dialog box was filled in to solve this problem, including every entry or change that needed to be made. 8Question #7: (10 points) Suppose someone were optimizing the food they ordered at a restaurant suchas Wendy’s with the twin goals of minimizing cost and minimizing fat in the food purchased, whilemeeting other relevant RDA requirements (e.g., getting enough protein, calories and vitamins; not gettingtoo much salt or cholesterol, etc.). Sketch the trade off curve that would be obtained by repeatedlysolving a goal programming version of this problem that sought to minimize the maximum weightedpercent deviation: a) Label both axes.b) Put a smiley face in the ideal cornerc) Draw the curve as a solid line with the appropriate slope and curvature.d) Label one side of the line as infeasible and the other side as suboptimal.e) Add a second dashed line showing the result if one repeated the exercise after relaxing thevitamins constraint.