90-722: Mgmt Science I, Midterm Solution, Morning Section, Spring 2017Question #1: (8 Points): On the first day of class, we created a 2D data table showing how the size of a loan payment depended on its term and interest rate. Show how the (“world’s worst”) dialog box was filled in to get Excel to complete the table.Solution:Row input cell “=A3” and Column input cell “=A2”.Question #2: (12 Points): a) How many decision variables and constraints (besides non-negativity) are there in a standard transshipment problem with 3 sources, 4 transshipment nodes, and 5 sinks, where every transshipment node is connected to every source and sink, but no source is connected directly to any sink? (Recall that source is a synonym for “supply node” and sink is a synonym for “destination” or “demand node”.) Answer: 32 decision variables and 12 constraints.b) In a standard make vs. buy problem, what would you expect to be the relationship between the objective function coefficients of the variables M3 and B3?1Answer: The coefficient for B3 would normally be greater. If it is cheaper to buy than to makesomething then there is no reason to make that product in house and you’d know ahead of time that M3* = 0 so product #3 could be dropped from the formulation.c) Suppose you solved a shortest path problem as a transshipment problem using the trick we discussed in class, shipping one unit at least cost from the origin to the destination. If the optimal solution had 7 decision variables whose value at optimality was 1, what would that tell you?Answer: The shortest path has 7 links. Question #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 value?b) The second constraint is X1 ≤ 3. What is its allowable decrease?c) What is the shadow price on constraint #2?d) Add to the diagram a constraint that would create degeneracy.e) What would the solution be if the objective function coefficient on X2 increased to 4?Answer:a) X* = (2, 4) so Z* = 6.b) The RHS can decrease by 1 before that constraint intersects the optimal solution.c) 0 since there is slack on that constraint. d) You can add any constraint that passes through the point (2,4).e) The solution would shift to the point (0, 5).2f) Question #4: (12 Points): Recall the Backpacks to Beyond (B2B) problems you worked for HW.Here is the spreadsheet after the optimization in HW #2, Problem #10.a) What type of problem is this?b) What is the solution and solution value?c) What constraints are binding?d) What does the 6 in Cell D10 mean?Solution:a) Product mixb) X1* = X2* = 0, X3* = 250, X4* = 125, and Z* = $22,500c) Material, stitching time, and non-negativity on Products X1 and X2.d) It takes 6 hours of labor to make one flashy backpack.Question #5: (20 Points): Formulate the LP in the previous problem.Solution:Letting the subscripts denote the backpacks types, i = 1, 2, 3, 4Maximize 30 X1 + 12 X2 + 50 X3 + 80 X4 s.t.X1 + X2 + X3 + X4 ≤ 37510 X1 + 20 X2 + 6 X3 + 11 X4 ≤ 4000X1 + 2 X3 + 4 X4 ≤ 1000Xi ≥ 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) What would the objective function coefficient for artisanal backpacks have to be in order for itto be optimal for B2B to make and sell some of them?3b) For what range of right-hand side values on the materials constraint would the nature of the solution remain the same?c) If B2B could acquire 100 hours of stitching time, how much should they be willing to pay for that additional resource?d) If B2B had to make one basic or one artisanal backpack, which would it rather make and why? a) If artisanal backpacks’ objective function coefficient rose by 8 to 20 then it could make some without losing money, and with any greater increase it would become optimal to start to make artisanal backpacks.b) 250 to 500c) Up to 100 * 15 = $1,500d) Basic, because its reduced cost is less unfavorable (only -$5 vs. -$8). Question #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 material available (i.e., right hand side of the first constraint).4a) What does the 50 under the Artisanal column mean?b) Why does it make sense that the first half dozen rows of the table are dominated by Flashy and Primo?c) What is the shadow price for material when there is between 525 and 550 units of it available?d) Why are the last six rows of the table all the same?Solution:a) If there are 550 units of material available then it is optimal to make 50 Artisanal backpacks.b) They produce the most profit per unit of material ($50 and $80 vs. $12 and $30 for X2 and X1) so when material is in relatively scarce supply, it should be used to make Flashy and Primo backpacks. c) Profits increased by $300 with that increase in 25 units of material so the shadow priceis $300 / 25 = $12 per unit of material.d) Once there are 575 units of material available, other resources become binding so further increases in the amount of material available do not change the optimal solution or solution value. Question #8: (9 Points): Draw the network diagram corresponding to this transshipment problem. 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 node.Minimize Z = X12 + 3 X13 + 5 X24 + 3 X25 + 5 X34 + 9 X35 + X45 Subject toX12 + X13 = 10X12  X24  X25 = 0X13  X34  X35 = 05X24 + X34  X45 = 0X25 + X35 + X45 = 10Xij ≥ 0 for all