Introduction to Management Science chap-3

8/20/2019 Introduction to Management Science chap-3

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Chapter 3 Linear Programming: Computer Solution and Sensitivity Analysis

1) The reduced cost (shadow price) for a positive decision variable is 0.

Answer: TRUE

2) hen the ri!ht"hand sides of 2 constraints are both increased b# 1 unit$ the value of the ob%ective function

will be ad%usted b# the su& of the constraints' prices.

Answer: A*E

+) hen a linear pro!ra&&in! proble& is solved usin! a co&puter pac,a!e decision variables will alwa#s be inte!er

and therefore decision variable values never need to be rounded.

Answer: A*E

-) *ensitivit# ran!es can be co&puted onl# for the ri!ht hand sides of constraints. Answer: A*E

) *ensitivit# anal#sis deter&ines how a chan!e in a para&eter affects the opti&al solution.

Answer: TRUE

/) The sensitivit# ran!e for an ob%ective function coefficient is the ran!e of values over which the current opti&al

solution point (product &i) will re&ain opti&al.Answer: TRUE

) The sensitivit# ran!e for an ob%ective function coefficient is the ran!e of values over which the profit does not

chan!e.

Answer: A*E

) The sensitivit# ran!e for a constraint 3uantit# value is the ran!e over which the shadow price is valid.

Answer: TRUE

4) 5f we chan!e the constraint 3uantit# to a value outside the sensitivit# ran!e for that constraint 3uantit#$ the shadow

price will chan!e.

Answer: TRUE

10) The sensitivit# ran!e for a constraint 3uantit# value is the ran!e over which the opti&al values of the

decision variables do not chan!e.

Answer: A*E

11)inear pro!ra&&in! proble&s are restricted to decisions in a sin!le ti&e period. Answer: A*E

12) A &ai&i6ation proble& &a# be characteri6ed b# all !reater than or e3ual to constraints.

Answer: A*E

1+) A chan!e in the value of an ob%ective function coefficient will alwa#s chan!e the value of the opti&al

solution.

Answer: A*E

1-) The ter&s reduced cost$ shadow price$ and dual price all &ean the sa&e thin!.

Answer: TRUE

1) *ensitivit# anal#sis can be used to deter&ine the effect on the solution for chan!in! several para&eters

at once.

Answer: A*E


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1/) or a profit &ai&i6ation proble&$ if the allowable increase for a coefficient in the ob%ective function is

infinite$ then profits are unbounded.

Answer: A*E

1) The reduced cost (shadow price) for a positive decision variable is .

Answer: 6ero

1) The sensitivit# ran!e for a is the ran!e of values over which the

3uantit# values can chan!e without chan!in! the shadow price

Answer: constraint 3uantit# 7iff: 2

14) is the anal#sis of the effect of para&eter chan!es on the opti&al solution.

Answer: *ensitivit# anal#sis

20) The sensitivit# ran!e for a constraint 3uantit# value is also the ran!e over which the

is valid.

Answer: shadow price21) The sensitivit# ran!e for an coefficient is the ran!e of

values over which the current opti&al solution point (product &i) will re&ain opti&al.

Answer: ob%ective function

8onsider the followin! linear pro!ra&$ which &ai&i6es profit for two products$ re!ular (R)$ and super (*):

9A 0R ; *

s.t.

1.2R ; 1./ * < /00 asse&bl# (hours) 0.R ; 0. * <

+00 paint (hours)

.1/R ; 0.- * < 100 inspection (hours)

Sensitivity Report:


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Final Reduced Objective Alloable AlloableCell !ame "alue Cost Coe##icient $ncrease %ecrease

=>= Re!ular ? 241./ 0.00 0 0 20=8= *uper ? 1++.++ 0.00 0 -+.

Final Shado Constraint Alloable AlloableCell !ame "alue Price R&'& Side $ncrease %ecrease

=E=+ Asse&bl# (hr@unit) /+.++ 0.00 /00 1E;+0 +/./=E=- aint (hr@unit) +00.00 ++.++ +00 +4.24 1

=E= 5nspect (hr@unit) 100.00 1-.+ 100 12.4- -0

22) The opti&al nu&ber of re!ular products to produce is $ and the opti&al nu&ber

of super products to produce is """""""" for the total profit of """""""""""""""""""

Answer: 241./$ 1++.++$ =2-$+

2+)5f the co&pan# wanted to increase the

available hours for one of their

constraints (asse&bl#$ paintin!$ or

inspection ) b# 2 hours$ the# should

increase .

Answer: 5nspection

2-) The profit on the super

product could increase b# without

affectin! the product &i.

Answer: =0.

2)5f downti&e reduced the available capacit# for paintin! b#

-0 hours (fro& +00 to 2/0 hours)$ profits would be reduced

b# .

Answer: =1$+++

2/) A chan!e in the &ar,et has

increased the profit on the super product b# =. Total

profit will increase b# .

Answer: =//


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Trac,saws$ 5nc. &a,es tractors and lawn &owers. The fir& &a,es a profit of =+0 on each tractor and =+0 on each lawn

&ower$ and the# sell all the# can produce. The ti&e re3uire&ents in the &achine shop$ fabrication$ and tractor asse&bl#

are !iven in the table.

or&ulation:

et ? nu&ber of tractors produced per period

# ? nu&ber of lawn &owers produced per period 9A +0 ;

+0#

sub%ect to 2 ; # < /0

2 ; +# < 120

< -

The !raphical solution is shown below.

2) Bow &an# tractors and saws should be produced to &ai&i6e profit$ and how &uch profit will the#

&a,eC

Answer: 1 tractors and +0 saws for =1$+0 in profit

2) 7eter&ine the sensitivit# ran!e for the profit for tractors.

Answer: 20 < < /0

24) hat is the shadow price for asse&bl#C

Answer: 0


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+0) hat is the shadow price for fabricationC

Answer: =.0

+1) hat is the &ai&u& a&ount a &ana!er would be willin! to pa# for one additional hour of &achinin!

ti&eC

Answer: =1+.0 " =1+0 ? =.0

+2) A brea,down in fabrication causes the available hours to drop fro& 120 to 40 hours. Bow will this

i&pact the opti&al nu&ber of tractors and &owers producedC

Answer: ? 22.$ # ? 1$ D ? 112$ so profits will fall b# =1+0 " =112 ? =22.

++) hat is the ran!e for the shadow price for asse&bl#C

Answer: allowable decrease ? - " 1 ? +0$ and allowable increase is .

The production manager for the Whoppy soft drink company is considering the production of 2kinds of soft drinks: regular (R) and diet (D). The company operates one "8 hour" shift per day.

Therefore the production time is !8 minutes per day. During the production process one of the

main ingredients syrup is limited to ma#imum production capacity of $%& gallons per day.

'roduction of a regular case reuires 2 minutes and & gallons of syrup hile production of a diet

case needs ! minutes and * gallons of syrup. 'ro+ts for regular soft drink are ,*. per case and

profits for diet soft drink are ,2. per case.

The for&ulation for this proble& is !iven below.

9A D ? =+R ; =27

s.t.

2R ; -7 < -0

R ; +7 < /

The sensitivit# report is !iven below


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+-) hat is the opti&al dail# profitC

Answer: =-20

+) Bow &an# cases of re!ular and how &an# cases of diet soft drin, should hopp# produce to &ai&i6e

dail# profitC

Answer: 40 cases of re!ular and cases of diet

+/) hat is the sensitivit# ran!e for the per case profit of a diet soft drin,C

Answer: 1. < c2 < /.0

+) hat is the sensitivit# ran!e of the production ti&eC Answer:

20 < b1 < 400

+) if the co&pan# decides to increase the a&ount of s#rup it uses durin! production of these soft drin,s to 440

lbs. will the current product &i chan!eC 5f show what is the i&pact on profitC Answer: Fes.$ 5ncrease in profit ?

0.(440 " /) ? =10

-allory furniture uys 2 products for resale: ig shel/es (0) and medium shel/es (-). 1ach ig shelf

costs ,& and reuires cuic feet of storage space and each medium shelf costs ,* and

reuires 3 cuic feet of storage space. The company has ,%& to in/est in shel/es this eek

and the arehouse has 8 cuic feet a/ailale for storage. 'rofit for each ig shelf is ,* and

for each medium shelf is ,&.4raphically sol/e this prolem and anser the folloing uestions.

+4) hat is the opti&al product &i and &ai&u& profitC

Answer: 10 bi! shelves and no &ediu& shelves. rofit ? =-$000


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-0) 7eter&ine the sensitivit# ran!e for the profit on the bi! shelf.

Answer: the ran!e is fro& =20 to infinit#

-1)5f the 9allor# urniture is able to increase the profit per &ediu& shelf to

=200$ would the co&pan# purchase &ediu& shelves. 5f so$ what would be the

new product &i and the total profitC

Answer: #es$ >i! ? 40$ 9ediu& ? 100 D ? =-$000

The linear pro!ra&&in! proble& whose output follows is used to deter&ine how &an# bottles of fire red nail polish (1)$

bri!ht red nail polish (2)$ basil !reen nail polish(+)$ and basic pin, nail polish(-) a beaut# salon should stoc,. The

ob%ective function &easures profitG it is assu&ed that ever# piece stoc,ed will be sold. 8onstraint 1 &easures displa# space

in units$ constraint 2 &easures ti&e to set up the displa# in &inutes. Hote that !reen nail polish does not re3uire an# ti&e

to prepare its displa#. 8onstraints + and - are &ar,etin! restrictions. 8onstraint + indicates that the &ai&u& de&and for

fire red and !reen polish is 2 bottles$ while constraint - specifies that the &ini&u& de&and for bri!ht red$ !reen and pin,

nail polish bottles co&bined is at least 0 bottles.

9A 1001 ; 1202 ; 10+ ; 12-

*ub%ect to 1. 1 ; 22 ; 2+ ; 2-


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Jb%ective 8oefficient Ran!es

Ri!ht Band *ide Ran!es

-2) Bow &uch space will be left unusedC Bow &an# &inutes of idle ti&e re&ainin! for settin! up the

displa#CAnswer: 0$ /+

-+) a) To what value can the per bottle profit on fire red nail polish drop before the solution

(product &i) would chan!eC

b) ># how &uch can the per bottle profit on !reen basil nail polish increase before the solution (product

&i) would chan!eC

Answer: a) .$ b) 12

--) a)># how &uch can the a&ount of space decrease before there is a chan!e in the profitC

b) ># how &uch can the a&ount of space decrease before there is a chan!e in the product &iC

c) ># how &uch can the a&ount of ti&e available to setup the displa# can increase before the solution (product

&i) would chan!eC

d) hat is the lowest value for the a&ount of ti&e available to setup the displa# before the solution (product

&i) would chan!eC

Answer: a) 0 b) c) 0 d)


9/17

-) Fou are offered the chance to obtain &ore space. The offer is for 1 units and the total price is =100.

hat should #ou doC h#C

Answer: re%ect the offer$ (11 1) ? 112 L 100

-/) 9a D ? 1 ; +2

*ub%ect to: /1 ; 22 < 1

11 ; 202 < /0

1 ; 2 I 0

7eter&ine the sensitivit# ran!e for each constraint.

Answer: constraint 1: / " 2-

constraint 2: - " 10

-) 9a D ? 1 ; +2

*ub%ect to: /1 ; 22 < 1

11 ; 202 < /0

1 ; 2 I 0

7eter&ine the sensitivit# ran!e for each ob%ective function coefficient. Answer: 1:

2.2 " 4.0 and$ 2: 1.// " /.//

-) 9a D ? +1 ; +2

*ub%ect to: 101 ; -2 < /0

21 ; 02 < 200

1 $ 2 I 0

7eter&ine the sensitivit# ran!e for each ob%ective function coefficient. Answer: 1:

1. " . and$ 2: 1.2 " /.0

-4) or a &ai&i6ation proble&$ assu&e that a constraint is bindin!. 5f the ori!inal a&ount of a resource is -

lbs.$ and the ran!e of feasibilit# (sensitivit# ran!e) for this constraint is fro& + lbs. to / lbs.$ increasin! the a&ount

of this resource b# 1 lb. will result in the:

A) sa&e product &i$ different total profit

>) different product &i$ sa&e total profit as before

8) sa&e product &i$ sa&e total profit

7) different product &i$ different total profit Answer:

A

0) A plant &ana!er is atte&ptin! to deter&ine the production schedule of various products to &ai&i6e

profit. Assu&e that a &achine hour constraint is bindin!. 5f the ori!inal a&ount of &achine hours available is 200

&inutes.$ and the ran!e of feasibilit# is fro& 1+0 &inutes to +-0 &inutes$ providin! two additional &achine hours

will result in:

A) the sa&e product &i$ different total profit


10/17

>) a different product &i$ sa&e total profit as before

8) the sa&e product &i$ sa&e total profit

7) a different product &i$ different total profit Answer: 7

The production &ana!er for >eer etc. produces 2 ,inds of beer: li!ht () and dar, (7). Two resources used to produce

beer are &alt and wheat. Be can obtain at &ost -00 o6 of &alt per wee, and at &ost +200 o6 of wheat per wee,

respectivel#. Each bottle of li!ht beer re3uires 12 o6 of &alt and - o6 of wheat$ while a bottle of dar, beer uses o6 of

&alt and o6 of wheat. rofits for li!ht beer are =2 per bottle$ and profits for dar, beer are =1 per bottle.

1)5f the production &ana!er decides to produce of 0 bottles of li!ht beer and -00 bottles of dar, beer$ it will result

in slac, of

A) &alt onl#

>) wheat onl#

8) both &alt and wheat

7) neither &alt nor wheat Answer:A

2) hich of the followin! is not a feasible solutionC

A) 0 and 0 7

>) 0 and -00 7

8) 200 and +00 7

7) -00 and -00 7

Answer: 7

What is the optimal eekly profit5

6) , 0) ,3 7) ,8 D) ,% 1) ,$

Answer: 8

9allor# urniture bu#s 2 products for resale: bi! shelves (>) and &ediu& shelves (9). Each bi! shelf costs =00 and

re3uires 100 cubic feet of stora!e space$ and each &ediu& shelf costs =+00 and re3uires 40 cubic feet of stora!e space. The

co&pan# has =000 to invest in shelves this wee,$ and the warehouse has 1000 cubic feet available for stora!e. rofit for

each bi! shelf is

=+00 and for each &ediu& shelf is =10.

+) hich of the followin! is not a feasible purchase co&binationC

A) 0 bi! shelves and 200 &ediu& shelves

>) 0 bi! shelves and 0 &ediu& shelves

8) 10 bi! shelves and 0 &ediu& shelves

7) 100 bi! shelves and 100 &ediu& shelves Answer:

7


11/17

-)5f the 9allor# urniture co&pan# decides to purchase 10 bi! shelves and no &ediu& shelves$ which of the

two resources will be left overC

A) invest&ent &one# onl#

>) stora!e space onl#

8) invest&ent &one# and stora!e space

7) neither invest&ent &one# nor stora!e space Answer: >

The production &ana!er for the hopp# soft drin, co&pan# is considerin! the production of 2 ,inds of soft drin,s: re!ular

and diet. The co&pan# operates one M hourM shift per da#. Therefore$ the production ti&e is -0 &inutes per da#. 7urin!

the production process$ one of the &ain in!redients$ s#rup is li&ited to &ai&u& production capacit# of / !allons per

da#. roduction of a re!ular case re3uires 2 &inutes and !allons of s#rup$ while production of a diet case needs - &inutes

and + !allons of s#rup. rofits for re!ular soft drin, are =+.00 per case and profits for diet soft drin, are =2.00 per case.

) hich of the followin! is not a feasible production co&binationC

a)3 R and %& D )*& R and D c) R and 2 D d)%& R and 3

D

e)& R and & D

Answer: 7

/) or the production co&bination of 1+ re!ular cases and 0 diet cases$ which resource is co&pletel#

used up (at capacit#)C

a) only time )only syrup c)time and syrup d)neither time nor syrup

6nser:0

) The sensitivit# ran!e for the profit on a re!ular case of soda is

A. =2 to =+ >.=2 to =- 8.=1 to =+ 7. =1 to =+.++

Answer:7

) hich of the followin! could not be a linear pro!ra&&in! proble& constraintC

A. A ; > < "+ >. A " > < "+ 8.A " > < + 7. A ; > I "+

E. "A ; > < "+

Answer: A

4) Use the constraints !iven below and deter&ine which of the followin! points is feasible.

(1) 1- ; /# < -2

(2) " # < +

6) # 9 y & 0) # 29 y 2 7) # 29 y 8 D) # 29 y ! 1) # *9 y .&


12/17

Answer: >

/0) or the constraints !iven below$ which point is in the feasible re!ion of this &ini&i6ation proble&C

) !#$y ; !2

2) ) the &ar!inal !ain in the ob%ective that would be reali6ed b# subtractin! 1 unit of a resource

8) the &ar!inal cost of addin! additional resources

7) the &ar!inal !ain of sellin! one &ore unit

Answer: A

/+) Niven the followin! linear pro!ra&&in! proble&:

9a D ? 1 ; 20 # s.t.

; # < -0 - ; # I -

hat would be the values of and # that will &ai&i6e revenueC

6) # &9 y 0) # 9 y 8 7) # 9 y D) # 9 y 1) # *9 y !

Answer: >

/-) Niven the followin! linear pro!ra& that &ai&i6es revenue:

9a D ? 1 ; 20 # s.t.

; # < -0 - ; # I -

/) hat is the &ai&u& revenue at the opti&al solutionC

6) ,2 0) ,$ 7) ,8& D) ,2

Answer: >

Niven the followin! linear pro!ra&&in! proble& that &ini&i6es cost. 9in D ? 2 ; #

*ub%ect to (1) ; -# I /-


13/17

(2) 2 ; -# I +2

(2) # I 2

//) 7eter&ine the opti&u& values for and #.

6) # 29 y $ 0) # $9 y 2 7) # 29 y 2 D) # 29 y 2 1) # $9 y

&

Answer: 8

/) At the opti&al solution the &ini&u& cost is:

6) ,* 0) ,! 7) ,& D) ,&2 1) ,&*.**

Answer: >

/) hat is the sensitivit# ran!e for the cost of C

a) to 2 ) ! to $ c) 2 to ! d) to !

) 6nser: D

/4) hat is the sensitivit# ran!e for the third constraint$ # I 2C

a) to ! ) 2 to &.** c) to &.** d) ! to $.**

Answer: 8

0) or a &ai&i6ation proble&$ the shadow price

&easures the in the value of the opti&al solution$ per unit increase

for a !iven .

a) decrease resource )increase parameter c)impro/ement resource d)change

o>ecti/e function coefficient e) decrease parameter

6nser: 7

1) *ensitivit# anal#sis is the anal#sis of the

effect of chan!es on the .

a)price company )cost production c)parameter optimal solution d)none of the ao/e

6nser: 7

2) or a linear pro!ra&&in! proble&$ assu&e that a !iven resource has not been full# used. e can

conclude that the shadow price associated with that constraint:

a)ill ha/e a positi/e /alue )ill ha/e a negati/e /alue c)ill ha/e a /alue of ?ero

d) could ha/e a positi/e negati/e or a /alue of ?ero. (no sign restrictions)

6nser: 7

+) or a resource constraint$ either its slac, value &ust be or its shadow price &ust be .

A) ne!ative$ ne!ative

>) ne!ative$ 6ero

8) 6ero$ 6ero

7) 6ero$ ne!ative

Answer: 8


14/17

Aunt Anastasia operates a s&all business: she produces seasonal cera&ic ob%ects to sell to tourists. or the sprin!$ she is plannin! to &a,e bas,ets$ e!!s$ and rabbits. >ased on #our discussion with #our aunt #ou construct the followin! table.

Four aunt also has co&&itted to &a,e 2 rabbits for a charitable or!ani6ation. >ased on the infor&ation in the table$ #oufor&ulate the proble& as a linear pro!ra&.

> ? nu&ber of bas,ets produced E ? nu&ber of

e!!s produced

R ? nu&ber of rabbits produced 9A 2.> ;

1.E ; 2R

s.t.

0. > ; 0.+++E ; 0.2R < 20 > ; E ; R < 0

0.2> ; 0.+++E ; 0.R < 0 R I 2

The Ecel solution and the answer and sensitivit# report are shown below.

(he Anser Report:


15/17

(he Sensitivity Report:

-) hich additional resources would #ou reco&&end that Aunt Anastasia tr# to obtainC

a) mi#@mold ) kilnc) paint and seal d)demand e)7annot tell from the

information pro/ided

6nser: 0

) *uppose the charitable or!ani6ation contacted Aunt Anastasia and told her that the# had overesti&ated the

a&ount of rabbits the# needed. 5nstead of 2 rabbits$ the# need +. Bow would this affect Aunt Anastasia's profitsC

a) 'rofits ould increase y ,&. 0)'rofits ould decrease y ,& c)'rofits ould

increase y ,2.&

c) 'rofits ould decrease y ,2.& e)7annot tell from the information pro/ided.Answer: >

/) Aunt Anastasia feels that her prices are too low$ particularl# for her e!!s. Bow &uch would her profit

have to increase on the e!!s before it is profitable for her to &a,e and sell e!!sC

6) ,.& 0) ,. 7) ,.& D) ,2.& 1) Aone of the ao/e

Answer: >

) Aunt Anastasia's available hours for paint and seal have fallen fro& 0 hours to /0 hours because of

other co&&it&ents. Bow will this affect her profitsC

6)'rofits ill decrease y ,*. 0) 'rofits ill increase y ,*. 7) 'rofits ill decrease y

,2.

D) 'rofits ill increase y ,2. 1) 'rofits ill not change.

Answer: E

) Aunt Anastasia can obtain an additional 10 hours of ,iln capacit# free of char!e fro& a friend. 5f she

did this$ how would her profits be affectedC


16/17

a)'rofit ould increase y ,2&. 0)'rofits ould decrease y ,2&. 7) 'rofits ould

increase y ,$.2&.

d) 'rofits ould decrease y ,$.2& e)7annot tell from the information pro/ided.

Answer: 8

4) Aunt Anastasia is plannin! for net sprin!$ and she is considerin! &a,in! onl# 2 products. >ased on the

results fro& the linear pro!ra&$ which two products would #ou reco&&end that she &a,eC

a) askets and eggs )askets and raits c)eggs and raits d)Bhe should

continue to make all *.

e) 7annot tell from the information pro/ided.

Answer: >

0illyCs 0lues sells * types of Tshirts: 6stro 0ling and 7urious. -anufacturing 6stros

reuires 2 minutes of machine time 2 minutes of laor and costs ,. 0rand 0ling reuires

2..& minutes of machine time * minutes of laor and costs ,! to produce. 0rand 7uriousreuires * minutes of machine time !& minutes of laor and costs ,8 to produce. There are *

machining hours a/ailale per eek *%& laor hours and he has a udget of ,*. 0rand

6stro sells for ,& 0rand 0ling for ,8 and 0rand 7urious for ,2&.

The for&ulation that &ai&i6es wee, profit shown below.

9A 1A ;1> ; 2 8

s.t.

2A ; 2.> ; +8 < +00

20A ; +0> ; -8 < +$0

10A ; 1-> ; 18 < +$000

The solution fro& O9 for indows is show below.


17/17

8) Ef 0illy could acuire more of any resource hich ould it e5a) machining time ) laor timec) money d) uyers

Answer: A

1)5f one of >ill#'s &achines brea,s down$ it usuall# results in about / hours of downti&e. hen this happens$

>ill#'s profits are reduced b#

6) ,& 0)8 7) ,2& D) ,*&

Answer: 7

2) >ill#'s accountant &ade an error$ and the bud!et has been reduced fro& =+000 to =200. >ill#'s profit

will !o down b#

6) , 0) ,$2& 7) ,*& D) ,$&

Answer: A

+) >ill# has decided that he can raise the price on the 8urious t"shirt b# 10P without losin! sales. 5f he raises the

price$ his profits will

a) increase y F )decrease y Fc) increase y ,2.& d) increase y

,2&

6nser: D

Introduction to Management Science chap-3

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