1

Name:___Solution_______

ESE 403

Operations Research

Fall 2010

Examination 1

Closed book/notes/homework/cellphone examination. You may use a calculator. Please

write on one side of the paper only. Extra pages will be supplied upon request.

You will only receive full credit if you show all your work.

Question Point value Your Score

1-5 15

6-10 15

11 15

12 15

13 15

14 25

Total 100

- -.:.: lIowing is not a necessary assumption before we can apply the Simplex

o 0 assumption."'"

?::;,xlJ"ti°onality assumption. V"

licativity assumption.o 0 ibility assumption. V-

I of the above are necessary ..eo

ints] For maximization problems, if the objective function evaluated at a Basic Feasible (BF~

solution is no smaller than its value at every adjacent BF solution, then

Qthe solution is optimal.

b. the solution is unbounded.

c. the problem is infeasible.

d. the problem has multiple optimal sol ti

e. None of the above is true.

3. (3 points) If a problem has two optimal Basic Feasible (BF) Solutions, then

a. it has infinitely many optimal BF solutions. .

b. it has infinitely many optimal ePE solutions.

Q it has infinitely many optimal solutions.

d. All of the above are true.

e. None of the above is true.

4. (3 points) The Simplex method a~s choose the entering basic variable that leads to

a. the best adjacent Basic Feasible (BF) solution.

-&.- the best adjacent objective function (largest zy"@ the direction of the maximum improvement._

d. All of the above are true.

e. None of the above is true.

5. (3 points) Simplex method's minimum ratio rule for choosing the leaving basic variable

a. always choose to remain in the feasible region as the entering variable increases.

b. always chooses to stop at the first constraint intersection as the entering variable

increases.

c. ~a~chooses the basic variable that will go to zero fir as entering yariable increases.

@ All of the above are true.

e. None of the above is true.

2

6. (3 points) In a particular iteration of the Simplex method, if there is a tie for which variable

should be the leaving basic variable, then the next BF solution

a. must not have any basic variable equal to zero.

b. must have at most two basic variables equal to zero.

c. must have exactly one basic variable equal to zero.

@ must have at least one basic variable equal to zero. / ./

e. must have infinitely many basic variables equal to zero. ~

7. (3 points) In the Simplex method, if there is no leaving basic variable at some iteration,

a. then the problem has no feasible solutions.

b. then the problem has multiple optimal solutions.

c. then the problem has exactly one optimal solution. J /d. then the problem has multiple unbounded optimal solutb/"

G) then the problem has unbounded optimal solution.

8. (3 points) When an artificial problem is created using the Big M method, if the basic solution of

any iteration contains an artificial variable

@ then the corresponding corner point solution is not feasible for th lglnal problem.

'b, then the corresponding corner point solution is optimal for e original problem.

~ then the corresponding corner point solution is unbo ed for the original problem.

X then the corresponding corner point solution is the single optimal solution."e... None of the above is true.

9. (3 points) In Simplex method, adjacent corner point solution of a problem with n decision

variable shares

n-l constraints.

a. n constraints.

c. n+l constraints.

d. n+2 constraints.

e. None of the above is true.

10. (3 points) In Simplex method, if we have n decisi variable and m constraints,

a. there will be n nonbasic variable in e basic feasible solution.

b. there will be m nonba ic varia in the basic feasible solution.

c. there will be n+m non ariable in the basic feasible solution.

d. there will be m*n nonbasic variable in the basic feasible solution.GJ None of the above is true. W\ \aQ~K V6..t-\o.b<.QS

VI - VV' V\ On b ((,S i"c, \fa,f"\-a\::f(o!>

3

11. (15 points) The professor in charge of ESE230 needs to schedule the staffing of the helpdesk.

The helpdesk opens from 8AM until midnight. From historical data of the demand, the following

minimum number oftutors are required to be on duty:

8AM-noon: 4 ®Noon-4PM: 8

4PM-8PM: 10

8PM-midnight: 6

Two types of tutors can be hired: graduate students and undergraduate students. Graduate

tutors work for 8 consecutive hours in any of the following shifts: morning (8AM-4PM),~

afternoon (noon-8PM), and evening (4PM-midnight). Graduate tutors are paid $20 an hour.----- ---Undergraduate tutors can be hired to work any of the four shifts listed above. Undergraduate

tutors are paid $10 an hour.

To make sure there is adequate supervision for every time period, there must be at least 2

graduate tutors on duty for every undergraduate tutor on duty.

The professor in charge would like to determine how many graduate and how many

undergraduate workers should work each shift to meet the requirement at the minimum

possible cost. Formulate an INTEGER linear programming model for this problem.

b = ji. o{ ~('"t\IA"+c. .c-tv.a en..\..l \ f\ I 3 AA-li p~ s~;x.fI

\2. PM - 6pf\ ..

;\>\ ~A.M.- 12-~M dtit+lI2.-?M - 4 PM ' "\.{'L: •.

4

Maximize

12. (15 points) Consider the following problem.

Subject to

Xl +2X2+ X3s 10

xl~2x2 +2X3~20

and

~~..k,.. torM ~(\~ ·vo.r\e.~ (~G.o.r~+",,\.o~ ')t-1c

1 1c~ '1c+- 1c:~ "K4- -:k-., ~\-\-fI'\~T"7 -:6 -\ -L.- I - \ D D 0

? """"'-"";;:LO ~ -:k-4 ~'7 \€.4,v.~C- ')(4- '(L; - \ 0 to "'l.

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13. (15 points) Consider the following problem.

Minimize Z=Xl-2x2

Subject toXl::; 4X2::;3

Xl+2x2 ~11

and

Solve using Simplex in tabular form. Show all steps.

kYtr hlwJ ~e ,.z.= &; -)) -iX~-;l.)

\)(1 - 2 .,x., ~ Lf

»:i-).+)(y<:~ ~

&1'-')) + j L '/;;.' -~) ., x:s ~X~·IIt

-- . x> \ -J} ')- 2 -z... 7\\ ~'J..1- ,..1

'X'\ 1- X) =- (,)(~ ~)(0j ~5 -)(,\ t-),(; - X'J t Y6 ::. I T

-)(;1 X~ X'11 Xs Yt')('(

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COI

-). 0 0 o M0.\ (j) V 0I 0 ) () 0) 0 {) '-1

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\ - fZ~~ X, XL! ~- )\& >(d enWSb

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--1 X4 lw-vc-5

~. 0 0 -\ \ It- >{J' eh4v-5o 0" -;f+JIA M ,~O \;1- 72;11\

0 \ . ,0 0 Q' c -/ ")(3 le--a\'-'0tL 0 I 0 0 ..-.

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0 B -;A -\ ~\- -:r:O--~'~-9~iMM-0 __ . ·~_e.........•__ ""-- ..•

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Minimize

14. (25 points) Consider the following problem.

subject to

2Xl + X2 ~ 10-3Xl + 2X2 ~ 6

Xl + X2 ~ 6and

a. Solve this problem graphically.

b. Using the Big M method, construct the complete first simplex tableau for the simplex

method and identify the corresponding initial (artificial)' BF solution. Also identify the

initial entering basic variable and the leaving basic variable.

c. Work through the Simplex method step by step to solve the problem. Make sure you

identify the entering basic variable, leaving basic variable, and the corresponding BF

solution.

d. What is the optimal solution value? What are the corresponding decision variable

values?

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