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90-722: Mgmt Science I, Midterm Solution, Afternoon Section, Spring 2017Question #1: (8 Points): I created a “division” table showing how various fractions convert to percentages. Show how the (“world’s worst”) dialog box was filled in to get Excel to complete that table.Solution:Row input cell “=A1” and Column input cell “=A2”.Question #2: (12 Points): a) How many decision variables and constraints (besides non-negativity) are there in a standard assignment problem that assigns 10 people to 10 tasks? Answer: 100 decision variables and 20 constraints.b) Consider a slight variation of the standard make vs. buy problem in which there are 4 constrained resources in one’s own factory and each of the five products can be made in-house or ordered from either of two distinct suppliers. How many decision variables and how many constraints (besides non-negativity) would there be?Answer: 15 decision variables (three for each of the 5 products) and 9 constraints (four resource constraints and five demand constraints).c) Suppose you solved a max flow problem as a transshipment problem using the trick we discussed in class, maximizing the flow from Node #1 to Node #20 in a network with 25 nodes and 100 arcs. How many decision variables and how many constraints (besides non-negativity and simple upper bounds) would there be in that formulation?Answer: 101 decision variables (including the artificial backflow that is the objective function)and 25 constraints. 1Question #3: (15 Points): I’ve pasted below the graphical solution to a standard two-variable product mix problem whose objective function is Max Z = X1 + X2. a) What is the optimal solution?b) Which constraints are binding?c) What would Z* be if the objective function coefficient on X1 increased to 3?d) Add to the diagram a redundant constraint.e) What is the reduced cost on variable X1? (in the original diagram)Answer:a) X* = (2, 4)b) Constraints #1 and #3c) The solution would shift to the point (3, 2) so Z* = 3 * 3 + 1 * 2 = 11.d) You can add any constraint that does not intersect the feasible region and that points back toward the feasible region. e) 0 since it is not forced up against an explicit lower or upper bound.Question #4: (12 Points): Recall that in HW #1, Problem #6, Howie’s friend Harold convinced Howie to commit to selling at least 45 “Super Bowl” kits that do-it-yourselfers (DIY) can assemble at home. I’ve pasted the spreadsheet from the optimization problem below.2a) What type of problem is this?b) What is the solution and solution value?c) What constraints are binding?d) What does the 12 in Cell B9 mean?Solution:a) Product mixb) X1* = 50, X2* = 0, X3* = 45, and Z* = $51,250c) Tubing, non-negativity on X2, and minimum production quantity on X3.d) It takes 12 feet of tubing to make one Aqua Spa. Question #5: (20 Points): Formulate the LP in the previous problem.Solution:Letting the subscripts denote the hot tub types, i = 1, 2, 3Maximize 350 X1 + 375 X2 + 750 X3 s.t.X1 + X2 + 2 X3 ≤ 1809 X1 + 6 X2 ≤ 150012 X1 + 16 X2 + 40 X3 ≤ 2400X3 ≥ 45Xi ≥ 0 for all i.Question #6: (12 Points): I’ve printed the sensitivity report for this problem below. Use it to answer these questions.a) For what range of objective function value coefficients on Aqua Spas is the current solution optimal?b) For what range of amounts of tubing would the nature of the solution remain the same?c) If Howie could acquire 100 more pumps, how much should they be willing to pay for that additional resource?3d) If Howie had to sell one Hydro Lux, what would the total profit be then? Solutiona) $281.25 and upb) 1800 to 2880c) $0d) $51,250 - $91.67 = $51,158.33Question #7: (12 Points): I ran a solver table to explore the implications of how the solution and solution value of this LP depends on the amount of tubing available.a) What does the 30 under the Hydrolux column mean?b) Why is the first line of the table blank?c) What is the shadow price for tubing when there is between 3000 and 3200 feet of tubing available?d) Write a sentence or two explaining how the number of Aqua Spas depends on the amountof tubing that is available and why that makes sense. [Write neatly. If your answer is hard to read, it won’t be graded.]4Solution:a) If Bob had 3,000 feet of tubing he should make 30 Hydroluxesb) There is no feasible solution if Howie only had 1,600 feet of tubing (because making 45 Super Bowls requires at least 45 * 40 = 1,800 feet of tubing). c) Profits increased by $1,250 with that increase in 200 feet of tubing so $1,250 / 200 = $6.25 per foot of tubing.d) The optimal number of Aqua Spas to make first increases but then decreases. That makes sense because if there is only 1,800 feet of tubing, it all has to go into making the required 45 Super Bowls. As more tubing becomes available it should go into making Aqua Spas because they produce the most profit per foot of tubing ($350 / 12 islarger than $375 / 16 or $750 / 40). But there is a relative abundance of tubing, start to phase out Aqua Spas in favor of Hydroluxes which produce more profit per pump. 5Question #8: (9 Points): Draw the network diagram corresponding to this transshipment problemthat is implemented in the node-arc incidence approach in Excel. Label arcs with their costs. Put the node’s ID number inside its circle and put the associated demand or supply (if any) in a box next to the

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