IDS 291



Sample Questions for Stevens’ Exam IICOB 291: Quantitative MethodsIMPORTANT! CHANGE YOUR VIEW IN THE RIBBON ABOVE TO DRAFT. (To see comments on the answer you select, point to it with your cursor. Try it here.)When we do RHS ranging, we study the effect of modifying the program bydefining new decision variableschoosing new values for the existing decision variableschoosing a new value for a constraint constantchanging the shadow price of a constraintchanging the profit contribution of a productWithin the RHS range of a constraint,the optimal objective function value is preservedthe optimal schedule is preservedthe optimal solution is preservedall shadow prices are preservedthe values of all slack variables remain unchangedA common problem requiring RHS ranging ischanging how much one unit of an activity contributes toward the program goalchanging the quantity available of a limited resourceremoving nonnegativity constraints from a programimposing a new constraintfinding the optimal tableauWhen we do OFCR ranging, we study the effect of modifying the program by, for example,defining new decision variableschoosing new values for the existing decision variableschoosing a new value for a constraint constantchanging the shadow price of a constraintchanging the profit contribution of a productWithin the OFCR range of a variable,the optimal objective function value is preservedthe optimal schedule is preservedthe optimal solution is preservedall shadow prices are preservednone of the above is necessarily true A problem likely to require OFCR ranging ischanging the amount of an activity that must be performed to meet a quota in a minimize cost problemchanging the quantity available of a limited resource in a minimize cost problemfinding the optimal tableauchanging the time required to complete an activity in a minimize time used problemintroducing a new decision variable into a maximize profit problemA certain linear program has two decision variables, x and y, and its goal is to maximize P. (P is some linear combination of x and y.) When the RHS of constraint 2 in this program is 100, the optimal solution is given by x = 4, y = 5, P = 100. When the RHS of this constraint is changed to 120, the optimal solution becomes x = 6, y = 25, P = 140. The RHS range of this constraint is 0 to 200. What is the shadow price of this constraint?SolutionQuestions 8-17 refer to the Circuit Village problem appearing on page 4 of this sample exam. Please read that page before proceeding. You may, if you wish, carefully remove the page from your exam for ease of reference. In the optimal solution, how many service contracts does Mark sell? AnswerIn what units is the slack in constraint 1 is measured? AnswerIn what units is the shadow price of constraint 5 is measured? AnswerThe value of the slack in constraint 6 has been deleted from the Excel report. What is its optimal value? AnswerState the RHS range of constraint 2. Include units. Answer Suppose Mark made as much money on a stereo sale as he now makes on a service contract sale. Compute, if possible, his new optimal income. AnswerSuppose Mark made as much money on a service contract sale as he now makes on a stereo sale. Compute, if possible, the effect of this change on Mark’s optimal income. AnswerMark agrees to work 4 hours of overtime today, in addition to his regular 8 hour day. Compute, if possible, Mark’s new optimal income. AnswerSuppose that Circuit Village has four wide screen TVs in stock, rather than the two mentioned. Determine, if possible, the effect of this change on Mark’s optimal income. AnswerThe OFCR range on TV extends to positive infinity. The objective of the program is to maximize Mark’s income. TV = the number of TVs Mark sells. Knowing these facts only about the Circuit Village problem, one could concludeif more TVs were available for sale, Mark could sell all that were availableif more TVs were available for sale, Mark couldn’t sell any additional TVs. Mark is selling all of the TVs he possibly can under the current circumstancesno feasible solution to this problem has Mark selling less TVs than he is now sellingno optimal solution to this problem has Mark selling less TVs than he is now sellingIn a particular linear program with the goal of MINIMIZE # of minutes required to grade exams, one of the constraints is M5 + M6 + M14 < 40. In English, this constraint says that the total number of multiple choice questions on the test (from Chapters 5, 6 and 14) cannot exceed 40. The shadow price of this constraint is -15. We are going to modify this limit of 40 on the number of multiple choice questions permitted. Assuming our change stays within the RHS range of this constraint, we can conclude no feasible solution to the program exists if the test is permitted to have more than 55 multiple choice questions.no optimal solution to the program exists if the test is permitted to have at most 25 multiple choice questions. the optimal solution to the problem as stated has 25 multiple choice questions on the exam.if the test could have up to 44 multiple choice questions, the required grading time would increase by an hour.if the test could have no more than 36 multiple choice questions, the required grading time would increase by an hour.Can you ever use sensitivity analysis to examine the effect of changing more than one objective function coefficient? AnswerCan you ever use sensitivity analysis to examine the effect of changing the constant term of more than one constraint? AnswerCan you ever use sensitivity analysis to examine the effect of changing both an objective function coefficient and a constraint constant? AnswerDefine GRAD to equal 1 if Bob is a graduate student, 0 if he is not. I want to write a constraint, acceptable to EXCEL, which says that Bob's grade, GRADE, must be at least 80 if Bob is a grad student, at least 70 if Bob is not. The constraint(s) saying this for Excel should be:GRADE >= 80 and GRADE >=70GRADE – 80 GRAD >= 0 and GRADE – 70 GRAD >= 0GRADE – 10 GRAD >= 70GRADE * GRAD >= 80 and GRADE * (1 – GRAD) >= 70GRADE * GRAD >= 150 23. If the constraint x < 100 has a slack of -20, what is the value of x? AnswerThe problem below is used for questions 8-17.Mark is the top salesman at Circuit Village, a consumer electronics store. His income is generated by his sales; he makes $3 on each stereo he sells, $99 on a large screen TV, $8 on each IBM computer, and $10 on a service contract. On average, it takes 10 minutes to sell a stereo or to convince a customer to take the service contract, 90 minutes to sell a TV, and 25 minutes to sell a computer. We assume that Mark is so persuasive that if he spends the time, he will make the sale. Mark works an 8-hour shift. Circuit Village requires that Mark sell at least 5 stereos and 5 service contracts a day. It has only 2 large screen TVs to sell. And obviously, service contracts are sold only in conjunction with some other sale. The LINDO report showing how Mark can maximize his income is shown below. If you wish, you may carefully remove this page from the rest of your test for easy reference.STEREO, SERVICE, TV, and IBM are the number of each of these four items that Mark sells. FRETIME is the number of minutes during his workday when he is not engaged in selling. Constraint 3 is more easily read asSERVICE <= STEREO + TV + IBM.MAX 3 STEREO + 10 SERVICE + 99 TV + 8 IBMSUBJECT TO1) 10 STEREO + 10 SERVICE + 90 TV + 25 IBM + FRETIME = 4802) SERVICE - STEREO - TV - IBM <= 03) SERVICE >= 54) TV <= 25) STEREO >= 5?Variables????????STEREOSERVICETVIBMFRETIME????14.0016.002.000.000.00????Constraints????LHS?RHS1: time101090251480=4802: no service alone-11-1-100<03: service quota0100016>54: TV limit001002<26: stereo quota1000014.00>5Profit3109980400.00=ProfitAdjustable Cells??FinalReducedObjectiveAllowableAllowableCellNameValueCostCoefficientIncreaseDecrease$B$3STEREO14.000.00372.71$C$3SERVICE16.000.0010116.33$D$3TV2.000.00991E+3044$E$3IBM0.00-4.7584.751E+30$F$3FRETIME0.00-0.6500.651E+30Constraints??FinalShadowConstraintAllowableAllowableCellNameValuePriceR.H. SideIncreaseDecrease$G$51: time LHS4800.654801E+30180$G$62: no service alone LHS03.501822$G$73: service quota LHS1605111E+30$G$84: TV limit LHS24421.82$G$96: stereo quota LHS14.000591E+30 Questions 24 -28 refer to the problem below.The student in this problem is deciding how much time to spend studying the two subjects of math and quant. She will be taking exams in each of these subjects, and wishes to maximize the total number of points she receives on the two exams combined. She can study math for no more than 7 hours, and her total study time cannot exceed 10 hours. She has decided that she must get at least 80 points on each of the exams. As in the pamphlet Quant/Math problem, the student will get a certain score on each of these tests without studying, and studying one and/or both subjects will improve both of her test scores.Suppose that we find the optimal solution to this problem, and discover that, in this solution, the total study time and quant test score constraints are binding, the math test score constraint is nonbinding, and the math study time constraint is redundant. Her optimal total test score is 170.Mark each of the following statements with either true or false. Note that in 25-28, you should choose true if the described result might happen. False on these questions means that the described consequence is impossible. All questions assume the student adopts the optimal solution. As always, when a statement involves changing one of the parameters of the problem, it is assumed that the other problem parameters given remain unchanged. In the optimal solution, the student gets a 90 on her math exam. AnswerIf the student were required to get at least 85 points on her quant test (rather than 80), her total exam score might improve. AnswerIf the student were required only to get at least 75 points on her math test (rather than 80), her total exam score might change. AnswerIf the number of hours that the student were permitted to study math were raised to 8, her total exam score might improve. AnswerIf the total number of hours that the students were permitted to study were raised to 11 (rather than 10), her total exam score might improve. Answer29. I have a MIN linear program that includes the constraint X >= 5. The RHS range of this constraint is 0 to 15, and its shadow price is 2. I want to know, if possible, what will happen to the optimal objective function value if I remove this constraint from the program. Determine the answer, if possible. AnswerWhat is the slack in x + 3y >= 4z? AnswerI have a 0/1 variable, TIMED, which is 1 if and only if the examination is timed. I want to write a constraint that say, "If the test is timed, spend at most 10 minutes total on problems 1, 2 and 3." Let MIN1, MIN2, and MIN3 be the number of minutes spent on questions 1, 2 and 3, respectively. Use the "big M trick" to represent this requirement. (Note: this may not be on the Fall 2002 test.. Check with Stevens.) AnswerQuestions 32 - 35 deal with the following situation. We have a linear program whose goal is to maximize profit in dollars. We have a constraint in this program, constraint number 2, which says that we cannot store more than 10 tons of our product. We have a variable in this program, LABOR, which measures the number of hours of unskilled labor we use. The shadow price of constraint 2 is measured inhours/tondollars/hourtonsdollars/tontons/dollarThe RHS range of constraint 2 is measured inhours/tondollars/hourtonsdollars/tontons/dollar The OFCR range of LABOR is measured inhours/dollardollars/hourhourshours/tontons/hour If the OFCR range of LABOR extends to negative infinity, thenthe program is unbounded.the program is infeasible.we could not feasibly use more labor than the optimal solution does.we could not feasibly use less labor than the optimal solution does.optimal profit would not change, even if labor became terrifically expensive. ................
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