1. Consider a linear program (LP): Min. X + 2Y st. – 2X + Y = 10 X + 3Y = k Y > 10 X, Y > 0 If the constant “h” is equal to which of the following, does the feasible region NOT exist? a. 23* b. 30 c. 37 d. 44 e. None of the others. 2. Which of the following would be an acceptable objective function for a linear programming model? Max. 7 X1 + Sin[X2]Min. X1 * X2Min. 2 X1 + X2 / 5*Min. 12 X12 – 17 X2None of the others. The following three questions are based on the following case: Dunder Miflin Technologies (DMT) produces trendy tablets. Two of its models are the market leaders: Supercool and Ubercool. Supercool has a profit margin of $130 and Ubercool brings in a profit of $180. Producing Supercool requires 1 processor and 3 standard memory chips whereas Ubercool needs 2 processors and 2 premium memory chips. DMT has 100 processors, 120 standard printing chips and 70 premium printing chips. Variable definitions for the linear programming problem, the profit maximization problem and the sensitivity analysis are provided below. You are required to answer the questions which follow using this information. Let, S = number of Supercool models produced U = number of Ubercool models produced Max. 130 S+ 180U Subject to: S + 2U < 100 (Processor Constraint) 3S < 120 (Standard Chip Constraint) 2U < 70 (Premium Chip Constraint) S, U > 0 FinalReducedObjectiveAllowableAllowableCellNameValueCostCoefficientIncreaseDecrease$A$2Supercool4001301E+3040$B$2Ubercool30018080180 FinalShadowConstraintAllowableAllowableCellNameValuePriceR.H. SideIncreaseDecrease$A$9Processor100901001060$A$10Standard Chip12013.33312018030$A$11Premium ChipXXXXXXXX701E+3010 3. What is the shadow (dual) price of the constraint “Premium Chip”? a. ∞ b. 90 c. 0 * d. 10 e. 60 4. If the objective function coefficient of “Supercool” is reduced by more than $50, which of the following could be the new optimal solution for “Ubercool”? a.34 b. 35* c. 36 d. 35 and 36 are possible. e. 34, 35 and 36 are all possible. 5. Suppose Sabre, a chip manufacturer, offers to sell 300 Standard Chips to DMT. At which of the following prices would DMT definitely be willing to purchase these 300 Standard Chips? I. $1,500 II. $2,000 III. $2,500 Only III and IIIOnly III, II and IIII and II* 6. Consider a partial output from a profit minimization problem that has been solved to optimality. FinalShadowConstraintAllowableAllowableNameValuePriceR.H. SideIncreaseDecreaseLabor Time700-3700300100 The Labor Time constraint is a resource availability constraint. What will happen to the dual value (shadow price) if the right-hand-side for this constraint decreases to 500? a. It will become a more negative number, such as -5.* b. It will become zero. c. It will become a positive number. d. It will become a less negative number, such as -2. e. None of the others. 7. The objective function of a maximization problem with 2 decision variables is -3x+7y. The optimal solution turns out to be x* =3 and y* =5. If the objective function becomes max -5x+7y, which of the following COULD be the new optimal solution for y? 5 and 6 are both possible solutions for y.*4 c. 5 d. 6 e. 4 and 5 are both possible solutions for y. Read the following case and answer the three questions that follow: A 400-meter medley relay involves four different swimmers, who successively swim 100 meters of the Backstroke, Breaststroke, Butterfly, and Freestyle. A coach has 4 very fast swimmers whose expected times (in seconds) in the individual events are given in the following table. The coach wants to assign the swimmers to events so that his chances of winning are maximized. Event SwimmerBreaststrokeBackstrokeButterflyFreestyleAlison80656055Beth65576552Carla75556350Denise70755854 Let Breaststroke be represented by 1, Backstroke by 2, Butterfly by 3 and Freestyle by 4 and the swimmers be referred to by the first alphabet of their names i.e. Alison is A etc. Let Xij represent swimmer “i”(A, B, C or D) assigned to event “j”(1,2,3 or 4). 8. The constraint for the Freestyle event can be written as: a. X14 + X24 + X34 + X44 < 1 b. X41 + X42 + X43 + X44 > 1 c. X14 + X24 + X34 + X44 = 1* d. X14 + X24 + X34 + X44 > 0 e. both c and d would work. 9. Using the Greedy Heuristic discussed in class, what would be the assignment of swimmers to events (note that this may NOT be the optimal solution)? a. A → 1, B → 2, C → 4, D → 3* b. A → 1, B → 2, C → 3, D → 4 c. A → 3, B → 2, C → 4, D → 1 d. A → 1, B → 3, C → 4, D → 2 e. A → 3, B → 1, C → 4, D → 2 10. If a swimmer is allowed to compete in more one event without loss of performance, then, in the optimal assignment, which swimmer would compete in more than one event? a. Alison b. Beth c. Carla* d. Denise e. No swimmer would compete in more than one event For the next six questions, consider the following: Burnside Marketing Research conducted a study for Barker Foods on some designs for a new dry cereal. Three attributes were found to be most influential in determining which cereal had the best taste: ratio of wheat to corn in the cereal flake, type of sweetener (sugar, honey, or artificial), and the presence or absence of flavor bits. Nine children participated in taste tests and provided the following part- worths for the attributes: Wheat/ CornSweetenerFlavor Bits Child Low High Sugar Honey Artificial Present Absent 1 18 17 13 17 15 11 21 2 15 20 20 27 25 24 11 3 25 16 10 15 20 7 10 4 15 10 10 26 20 18 10 5 10 10 15 20 30 16 11 6 15 17 19 16 15 11 13 7 19 12 19 16 9 14 10 8 18 12 14 7 20 15 8 9 10 25 20 10 15 8 6 Assume that the overall utility (sum of part-worths) of the current favorite cereal is 50 for each child. Your job is to design a product that will maximize the share of choices for the nine children in the sample. 11. Suppose that the optimal solution indicates a cereal with high wheat/corn ratio, sugar and no flavor bits. Then, Child #9 will prefer this optimal cereal. a. True* b. False 12. Suppose Child #4 prefers a particular solution. Then, which of the following children will also DEFINITELY prefer this solution? Child #7 b. Child #1 c. Child #3 d. Child #2* e. None of the others 13. Which child would definitely not switch from his/her current favorite cereal? a. Child #3 b. Child #5 c. Child #6* d. Child #7 e. None of the others. 14. If Child #7 switches from his/her current favorite cereal for a particular combination, how many children would NOT switch? a. 3 b. 4 c. 5 d. 6 e. 7* 15. If Child #2 does NOT switch from his/her current favorite cereal for a particular combination, how many children would actually switch? a. 4 b. 3 c. 2 d. 1* e. 0 Answer the next three questions based on the case given below: Tower Engineering Corporation is considering undertaking several proposed projects for the next fiscal year. The projects, the number of engineers and the number of support personnel required for each project, and the cost for each project are summarized in the following table: Projects123456Engineers Required503540455530Cost ($1,000,000s)2.03.42.62.83.02.5 Formulate an integer linear program that minimizes Tower’s cost, subject to constraints, which will be stated in the questions that follow. Let Pi = 0 or 1, indicate if project i will not be undertaken or will be undertaken respectively. 16. The most appropriate objective function for Tower Engineering Corporation is given as: a. Max. 2P1 + 3.4P2 + 2.6P3 + 2.8P4 + 3P5 + 2.5P6 b. Min. 50P1 + 35P2 + 40P3 + 45P4 + 55P5 + 30P6 c. Min. 2P1 + 3.4P2 + 2.6P3 + 2.8P4 + 3P5 + 2.5P6* d. Max. 50P1 + 35P2 + 40P3 + 45P4 + 55P5 + 30P6 17. Project 1 can be completed only if at least two out of Projects 2, 3 and 4 are completed first. The constraint that captures this is: a. P1 < P2 + P3 + P4 b. 2P1 < P2 + P3 + P4* c. 3P1 < P2 + P3 + P4 d. P1 < P2 + P3 + P4 – 1 e. b and c would both work. 18. If Project 5 is not completed, then at most one out of Projects 2, 4 and 6 can be completed. This requirement is represented by: a. 2P5 + 1 > P2 + P4 + P6* b. 2P5 + 1 < P2 + P4 + P6 c. 2P5 > P2 + P4 + P6 d. 3P5 > P2 + P4 + P6 e. None of the others. 19. The 2-D (i.e. (x,y)) graph of a problem that requires x to be an integer between 8 and 19 (inclusive on both ends) and y to be an integer between 265 and 288 (inclusive on both ends) has a feasible region _________. a. of 288 dots.* b. of two horizontal stripes. c. of two vertical stripes. d. of 253 dots. e. of 36 dots. Answer the next three questions based on the case given below: Hansen Controls has been awarded a contract for a large number of control panels. To meet this demand, it will use its existing plants in Houston and Tulsa, and consider new plants in San Diego, St. Louis, and Portland. Finished control panels are to be shipped to Seattle, Denver, and Kansas City. Pertinent information is given in the table. SourcesConstruction CostShipping Cost to Destination:CapacitySeattle 1Denver 2Kansas City 3Tulsa—-12635,000Houston—-10866,000San Diego800,00057820,000St. Louis400,00012427,000Portland450,0004101115,000Demand14,00011,0008,000 We develop a transportation model as an LP that includes provisions for the fixed costs (construction costs in this case) for the three new plants. The solution of this model would reveal which plants to build and the optimal shipping schedule. Letxij = the number of panels shipped from source i to destination jyi = 1 if plant i is built, = 0 otherwise (i = 3, 4, 5) 20. The constraint for supply from Portland is given as: a. x51 + x52 + x53 < 15,000 b. 4x51 + 10x52 + 11x53 < 15,000*y5 c. x15 + x25 + x35 < 15,000*y5 d. x31 + x32 + x33 < 20,000*y3 e. None of the others.* 21. The constraint for demand at Seattle is given as: a. x11 + x21 + x31 + x41 + x51 > 14,000* b. x11 + x21 + x31 + x41 + x51 < 14,000 c. x11 + x21 + x31 + x41 + x51 = 14,000 y1 d. x11 + x21 + x31 + x41 + x51 > 14,000 y1 e. x11 + x12 + x13 + x14 + x15 > 14,000 22. Suppose that Hansen has a budget crisis and therefore cannot build any new plants; they are left with only Tulsa and Houston plants. Assume that they’d want to meet as much of the demand at minimum cost. If Hansen applies the low-cost algorithm (also known as Greedy heuristic) we talked about in class to allocate demand, which city gets some but not all of its demand satisfied? a. Seattle b. Kansas City c. Denver* d. Denver and Kansas City both Read the following case and answer the three questions that follow: A University Police Department had decided to install emergency telephones at select locations on campus. The department wants to install the minimum number of telephones provided that each of the main campus streets is served by at least one telephone. The figure below maps the principal streets (A to K) on campus. It is logical to place the telephones at intersections of streets so that each telephone will serve at least two streets. For example, Location 2 provides coverage for Streets F, G and H according to the figure below, which shows the layout of the streets and the telephone locations (encircled numbers). 23. The constraint x3 + x5 ≥ 1 represents the constraint for: a. Street I b. Street J c. Street A d. Street K e. None of the others. 24. Which pair of locations, if they become unavailable, would make the problem infeasible? a. 1 and 4 b. 2 and 3* c. 5 and 6 d. 4 and 8 e. Both b and c would be correct. 25. Using the Greedy Heuristic Approach, the first two phones would be placed, respectively, at: a. Location 3 and then 4 b. Location 4 and then 7 c. Location 3 and then 7* d. Location 2 and then 3 e. (a) and (b) would both work. 26. Consider an all-binary problem with 5 variables and 8 constraints, excluding the non-negativity ones. The number of feasible solutions to this problem CANNOT be: a. 0 b. 10 c. 25 d. 35* e. Any of the above could be the number of feasible solutions. Please answer the next two questions based on the case below (Assume that the terminal values are payoffs. Please fill in any missing probability values as needed.) 27. Referring to Figure 1: What is the optimal decision based upon the Expected Value criterion? a. Plan A* b. Plan B c. Both are optimal 28. What is the Expected Value of Perfect Information (EVPI)? a. $2,200* b $8,800 c. $500 d. $2,700 e. None of the others. The next two questions refer to the following case: I sell snowblowers. They are to be ordered several months before the winter. Any units unsold at the end of the winter will have to be discarded .The unit cost is $35 and the selling price is $100/unit. There is no restriction on the lot size, in short, you can order 363 units, 474 units etc. i.e. in any denomination. 29. To maximize expected profit how many units should I order at the beginning of the season if I am allowed to order only once? Demand is Uniform between 300 and 800. 400455538625*685 30. If the demand were Normally distributed with mean of 600 and variance 3600, what would be my optimal order quantity? a. 648 b 651 c. 624* d. 646 e. 674 31. The material DX is used uniformly throughout the year. The data about annual requirement, ordering cost and holding cost of this material is given below: Annual requirement: 2,400 unitsOrdering cost: $10 per orderHolding cost: $0.30 per unit Determine the economic order quantity (EOQ) of material DX using above data. a. 400* b. 16,000 c. 300 d. 600 e. 200 32. In an EOQ problem, if the demand doubles, then the optimal order quantity would increase by (approximately): a. 41%* b. 100% c. 50% d. 200% e. 25% 33. Which of the following combinations guarantee that any local minimum is also a global minimum for a non-integer problem? a. Concave objective function and concave feasible region b. Concave objective function and convex feasible region c. Convex objective function and concave feasible region d. Convex objective function and convex feasible region* The next two questions refer to the following case: Skooters Skateboards produces two models of skateboards, the R and the M. Skateboard revenue (in $l,000s) for the firm is nonlinear and is stated as: (6R – 0.25R2) + (8M – 0.5M2). Skooters has 100 labor-hours available per week in its paint shop. Each R requires 5 labor-hours to paint and each M requires 6 labor-hours. The Management Science team at Skooters formulates its production planning problem to determine how many R and M skateboards should be produced per week in order to maximize revenue. Let R = number of R skateboards to produce per week M = number of M skateboards to produce per week 34. The solution to the unconstrained non-linear programming problem is R* = 12 and M* = 8. What is the minimum number of additional hours of labor that Skooters needs per week to make this solution also optimal for the constrained problem? a. 4 b. 8* c. 16 d. 32 e. 0 35. Which of the following can NOT be the objective function value of the constrained maximization problem (assume 100 hours of labor are available)? a. 69,000. * b. 65,000. c. 55,000. d. 62,000. e. All of the above could be the objective function value of the constrained problem.
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