Sample Questions for Stevens’ Exam II

COB 291: Quantitative Methods

IMPORTANT! 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[SPS1].)

  1. When we do RHS ranging, we study the effect of modifying the program by

a)defining new decision variables[SPS2]

b)choosing new values for the existing decision variables[SPS3]

c)choosing a new value for a constraint constant[SPS4]

d)changing the shadow price of a constraint[SPS5]

e)changing the profit contribution of a product[SPS6]

  1. Within the RHS range of a constraint,

a)the optimal objective function value is preserved[SPS7]

b)the optimal schedule is preserved[SPS8]

c)the optimal solution is preserved[SPS9]

d)all shadow prices are preserved[SPS10]

e)the values of all slack variables remain unchanged[SPS11]

  1. A common problem requiring RHS ranging is

a)changing how much one unit of an activity contributes toward the program goal[SPS12]

b)changing the quantity available of a limited resource[SPS13]

c)removing nonnegativity constraints from a program[SPS14]

d)imposing a new constraint[SPS15]

e)finding the optimal tableau[SPS16]

  1. When we do OFCR ranging, we study the effect of modifying the program by, for example,

a)defining new decision variables[SPS17]

b)choosing new values for the existing decision variables[SPS18]

c)choosing a new value for a constraint constant[SPS19]

d)changing the shadow price of a constraint[SPS20]

e)changing the profit contribution of a product[SPS21]

  1. Within the OFCR range of a variable,

a)the optimal objective function value is preserved[SPS22]

b)the optimal schedule is preserved[SPS23]

c)the optimal solution is preserved[SPS24]

d)all shadow prices are preserved[SPS25]

e)none of the above is necessarily true[SPS26]

  1. A problem likely to require OFCR ranging is

a)changing the amount of an activity that must be performed to meet a quota in a minimize cost problem[SPS27]

b)changing the quantity available of a limited resource in a minimize cost problem[SPS28]

c)finding the optimal tableau[SPS29]

d)changing the time required to complete an activity in a minimize timeused problem[SPS30]

e)introducing a new decision variable into a maximize profit problem[SPS31]

  1. A 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?

Solution[SPS32]

Questions 8-17 refer to the CircuitVillage 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.

  1. In the optimal solution, how many service contracts does Mark sell? Answer[SPS33]
  1. In what units is the slack in constraint 1 is measured? Answer[SPS34]
  2. In what units is the shadow price of constraint 5 is measured? Answer[SPS35]
  3. The value of the slack in constraint 6 has been deleted from the Excel report. What is its optimal value? Answer[SPS36]
  4. State the RHS range of constraint 2. Include units. Answer[SPS37]
  5. 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. Answer[SPS38]
  6. Suppose 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. Answer[SPS39]
  7. Mark agrees to work 4 hours of overtime today, in addition to his regular 8 hour day. Compute, if possible, Mark’s new optimal income. Answer[SPS40]
  8. Suppose that CircuitVillage 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. Answer[SPS41]
  9. The 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 CircuitVillage problem, one could conclude

a)if more TVs were available for sale, Mark could sell all that were available[SPS42]

b)if more TVs were available for sale, Mark couldn’t sell any additional TVs. [SPS43]

c)Mark is selling all of the TVs he possibly can under the current circumstances[SPS44]

d)no feasible solution to this problem has Mark selling less TVs than he is now selling[SPS45]

e)no optimal solution to this problem has Mark selling less TVs than he is now selling[SPS46]

  1. In 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

a)no feasible solution to the program exists if the test is permitted to have more than 55 multiple choice questions.[SPS47]

b)no optimal solution to the program exists if the test is permitted to have at most 25 multiple choice questions. [SPS48]

c)the optimal solution to the problem as stated has 25 multiple choice questions on the exam.[SPS49]

d)if the test could have up to 44 multiple choice questions, the required grading time would increase by an hour.[SPS50]

e)if the test could have no more than 36 multiple choice questions, the required grading time would increase by an hour.[SPS51]

  1. Can you ever use sensitivity analysis to examine the effect of changing more than one objective function coefficient? Answer[SPS52]
  2. Can you ever use sensitivity analysis to examine the effect of changing the constant term of more than one constraint? Answer[SPS53]
  3. Can you ever use sensitivity analysis to examine the effect of changing both an objective function coefficient and a constraint constant? Answer[SPS54]
  4. Define 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:

a)GRADE >= 80 and GRADE >=70[SPS55]

b)GRADE – 80 GRAD >= 0 and GRADE – 70 GRAD >= 0[SPS56]

c)GRADE – 10 GRAD >= 70[SPS57]

d)GRADE * GRAD >= 80 and GRADE * (1 – GRAD) >= 70[SPS58]

e)GRADE * GRAD >= 150 [SPS59]

23. If the constraint x 100 has a slack of -20, what is the value of x? Answer[SPS60]

The problem below is used for questions 8-17.

Mark is the top salesman at CircuitVillage, 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. CircuitVillage 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 as

SERVICE <= STEREO + TV + IBM.

MAX 3 STEREO + 10 SERVICE + 99 TV + 8 IBM

SUBJECT TO

1) 10 STEREO + 10 SERVICE + 90 TV + 25 IBM + FRETIME = 480

2) SERVICE - STEREO - TV - IBM <= 0

3) SERVICE >= 5

4) TV <= 2

5) STEREO >= 5

Variables
STEREO / SERVICE / TV / IBM / FRETIME
14.00 / 16.00 / 2.00 / 0.00 / 0.00
Constraints / LHS / RHS
1: time / 10 / 10 / 90 / 25 / 1 / 480 / = / 480
2: no service alone / -1 / 1 / -1 / -1 / 0 / 0 / 0
3: service quota / 0 / 1 / 0 / 0 / 0 / 16 / 5
4: TV limit / 0 / 0 / 1 / 0 / 0 / 2 / 2
6: stereo quota / 1 / 0 / 0 / 0 / 0 / 14.00 / 5
Profit / 3 / 10 / 99 / 8 / 0 / 400.00 / = / Profit
Adjustable Cells
Final / Reduced / Objective / Allowable / Allowable
Cell / Name / Value / Cost / Coefficient / Increase / Decrease
$B$3 / STEREO / 14.00 / 0.00 / 3 / 7 / 2.71
$C$3 / SERVICE / 16.00 / 0.00 / 10 / 11 / 6.33
$D$3 / TV / 2.00 / 0.00 / 99 / 1E+30 / 44
$E$3 / IBM / 0.00 / -4.75 / 8 / 4.75 / 1E+30
$F$3 / FRETIME / 0.00 / -0.65 / 0 / 0.65 / 1E+30
Constraints
Final / Shadow / Constraint / Allowable / Allowable
Cell / Name / Value / Price / R.H. Side / Increase / Decrease
$G$5 / 1: time LHS / 480 / 0.65 / 480 / 1E+30 / 180
$G$6 / 2: no service alone LHS / 0 / 3.5 / 0 / 18 / 22
$G$7 / 3: service quota LHS / 16 / 0 / 5 / 11 / 1E+30
$G$8 / 4: TV limit LHS / 2 / 44 / 2 / 1.8 / 2
$G$9 / 6: stereo quota LHS / 14.00 / 0 / 5 / 9 / 1E+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 studytime 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.

  1. In the optimal solution, the student gets a 90 on her math exam. Answer[SPS61]
  1. If the student were required to get at least 85 points on her quant test (rather than 80), her total exam score might improve. Answer[SPS62]
  2. If the student were required only to get at least 75 points on her math test (rather than 80), her total exam score might change. Answer[SPS63]
  3. If the number of hours that the student were permitted to study math were raised to 8, her total exam score might improve. Answer[SPS64]
  4. If 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. Answer[SPS65]

29. 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. Answer[SPS66]

  1. What is the slack in x + 3y >= 4z? Answer[SPS67]
  1. I 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.) Answer[SPS68]

Questions 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.

  1. The shadow price of constraint 2 is measured in

a) hours/ton

b) dollars/hour

c) tons[SPS69]

d) dollars/ton[SPS70]

e) tons/dollar[SPS71]

  1. The RHS range of constraint 2 is measured in

a) hours/ton

b) dollars/hour[SPS72]

c) tons[SPS73]

d) dollars/ton

e) tons/dollar[SPS74]

  1. The OFCR range of LABOR is measured in

a) hours/dollar[SPS75]

b) dollars/hour[SPS76]

c) hours

d) hours/ton

e) tons/hour[SPS77]

  1. If the OFCR range of LABOR extends to negative infinity, then

a) the program is unbounded.

b) the program is infeasible.[SPS78]

c) we could not feasibly use more labor than the optimal solution does.[SPS79]

d) we could not feasibly use less labor than the optimal solution does.[SPS80]

e) optimal profit would not change, even if labor became terrifically expensive.[SPS81]

[SPS1]1That's the idea!

[SPS2]1No.

[SPS3]1We do not choose the optimal values—LINDO finds them.

[SPS4]1Yes. This constant is usually on the RHS.

[SPS5]1No. We don’t control shadow price in our formulation.

[SPS6]1No. This is OFCR ranging.

[SPS7]1No. In general, no kind of ranging preserves this.

[SPS8]1No. That’s OFCR ranging

[SPS9]1No. Objective function value is part of the solution., and answer a was wrong!

[SPS10]1Correct. So is the linear relation between the RHS value and ALL decision and slack varables.

[SPS11]1No. When the schedule changes, the slacks will generally change.

[SPS12]1No. This is OFCR ranging.

[SPS13]1Yes. RHSs of constraints are frequently available quantities.

[SPS14]1No—LINDO doesn’t print out RHS ranges for nonnegativity constraints.

[SPS15]1No. If the new constraint were violated by the old optimal solution, we couldn’t address such a change at all. If the new constraint were satisfied by the old optimal solution, the optimal solution would stay the same. There is no relevance of RHS ranging to this problem.

[SPS16]1No. We’re not even talking about optimal tableaux this semester!

[SPS17]1No.

[SPS18]1No. We don’t choose the optimal values—LINDO finds them.

[SPS19]1No. This is RHS ranging.

[SPS20]1No. We don’t control this directly.

[SPS21]1Yes. A common objective is to maximize profit. In such problems, OFCR is changing the profit per unit of something.

[SPS22]1No. If some activity that you are doing changes its worth, then your optimal objective value probably changes, too.

[SPS23]1Yes. What you are making or doing doesn’t change.

[SPS24]1No. Optimal solution includes the optimal objective value, and a) was wrong!

[SPS25]1No. That’s RHS ranging.

[SPS26]1Nope…one of them is right.

[SPS27]1NO…that’s a RHS change.

[SPS28]1No. That’s a RHS change.

[SPS29]1We’re not even doing this this semester!

[SPS30]1Yes. OFCR always changes the worth per unit of something, where worth means “whatever the objective is measuring”. Here, worth is in minutes of time.

[SPS31]1Nope…this is an “introduction of a new decision variable”problem.

[SPS32]1Only three points are relevant to this problem:

1.Are both RHS values in the RHS range. Answer: Yes.

2.By how much does the RHS change? Answer: 20

3.When this change occurs, by how much does the objective change? Answer: +40

The shadow price is then the amount the objective improves per unit change in the RHS. Since the objective increases by 40 when the RHS increases by 20, the objective changes by +2 each time the RHS increases by 1, so the shadow price is +2. (Note that, for a MIN, the answer would be –2. Note, too, that if the RHS = 200, the optimal solution would be x = 14, y = 25, and P = 300.)

[SPS33]116, the optimal value of SERVICE

[SPS34]1The same units as the constraint itself: minutes. Note that since the constraint is an equality, this slack is equal to 0.

[SPS35]1Objective function units per constraint units: $/stereo.

[SPS36]1Since STEREO = 14 in the optimal solution, the optimal value of the slack in constraint 6 is 14-5 = 9 stereos. (Use definition of slack.)

[SPS37]1From the LINDO report, this is –22 to 18 contracts.

[SPS38]1OFCR on STEREO extends up to $10, the proposed change is within the OFCR range (barely), and optimal schedule does not change. Income therefore increases by 7(14) = $98.

[SPS39]1OFCR on SERVICE does not allow a $7 decrease. The proposed change is outside of the OFCR range, so we cannot compute the new income, since optimal schedule may change.

[SPS40]1RHS on constraint 2 allows an infinite increase, so the 240 minute increase is within the range, and shadow prices hold. Income increases by 240  .65 = $156.00.

[SPS41]1RHS on 5) does not allow an increase of 2, so shadow prices probably change, and we cannot answer the question.

[SPS42]1Certainly untrue—he has limited time for sales.

[SPS43]1We would not be able to determine the truth of this statement from the information in this problem alone. In the original problem, this statement is certainly untrue, since the RHS range of 5) allows an increase of more than 1.

[SPS44]1Yes. If you don't understand this, reread the section in your pamphlet about OFCR ranges that extend to +.

[SPS45]1Backward. Clearly, in the original problem, Mark would be allowed to modify his current solution by selling no TVs, so this answer can't be right.

[SPS46]1Whether this statement is true or not cannot be ascertained from the information in this problem alone. The statement happens to be true for the original problem, but that is irrelevant. If for example, a TV sold for $55, then the optimal solution given in the original problem would still be optimal , but alternative optima selling fewer TVs would exist.

[SPS47]1No. If the change is within the RHS range, the shadow prices are fixed, so certainly the program stays feasible!

[SPS48]1No. Within the RHS range, shadow prices hold, and this only makes sense if the the program still has an optimal solution.

[SPS49]1No. In fact, since the shadow price of the constraint is nonzero, we know that the constraint is binding in the current optimal solution. The current optimal solution thus has 40 multiple choice questions.

[SPS50]1Almost, but not quite. If you use the three sign rule, you'll see this answer is backward. Why? Because the limit is on the number of multiple choice questions that the test is allowed to have. You make up a test with at most 40 multiple choice questions, doing the best you can. Now I tell you, "Oh, you can have up to 44, if you want." If it helps you, great. If not, stick with what you've got. Giving you more flexibility can't make your answer get worse.

To give you an idea of how such a situation could arise, let me give you an example. I have to make a 100 question test. I want to grade it as fast as I can. If I'm allowed to have more multiple choice questions, that may mean less essay questions, saving time.

[SPS51]1Correct. See the answer to d) if you guessed at this answer.

[SPS52]1Yes…you need to use the 100% rule for OFCR ranging.

[SPS53]1Yes. You need to use the 100% rule for RHS ranging.

[SPS54]1No. You can't combine RHS and OFCR ranging. What would be preserved?

[SPS55]1These two constraint would force GRADE to be at least 80, no matter what GRAD is. (All constraints in a LINDO program must be satisfied.)

[SPS56]1If GRAD = 0, this requires only that GRADE be at least 0. Not good enough.