Queuing: Waiting-Line Models Problem 1:
A new shopping mall is considering setting up an information desk manned by one employee. Based upon information obtained from similar information desks, it is believed that people will arrive at the desk at a rate of o f 20 per hour. It takes an average of 2 minutes to answer a uestion. It is assumed that the arrivals follow a !oisson distribution and answer times are e"ponentially distributed. #a$ %ind the the probabili probability ty that that the employee employee is idle. idle. #b$ %ind the the proportion proportion of the the time that that the employee employee is busy busy.. #c$ %ind the average average number number of people receiving receiving and and waiting to to receive some some information information.. #d$ %ind the average average number number of people waiting waiting in line line to get some informati information. on. #e$ %ind the the average time time a person person seeking seeking informatio information n spends in in the system. system. #f$ %ind the e"pected e"pected time time a person spends spends &ust waiting waiting in line line to have a uestion uestion answered answered #time in the ueue$. Problem 2:
Assume that the information desk employee in !roblem ' earns ('0 per hour. )he cost of waiting time, in terms of customer unhappiness with the mall, is ('2 per hour of time spent waiting in line. %ind the total e"pected costs over an *+hour day. Problem 3:
)he shopping mall has decided to investigate the use of two employees on the information desk. #a$ %ind the the probabili probability ty of no people people in the the system. system. #b$ %ind the the average average number number of people people waiting waiting in in this system. system. #c$ %ind the the e"pected e"pected time time a person person spends spends waiting waiting in this this system. system. #d$ Assuming Assuming the same salary salary level and waiting waiting costs as in !roblem !roblem 2, find the total e"pected e"pected costs costs over an *+hour day.
'
Problem 4:
)hree students arrive per minute at a coffee machine that dispenses e"actly four cups per minute at a constant rate. escribe the system parameters. Problem 5:
A repairman repairman at a local metal working shop services their five drill presses. -ervice time averages '0 minutes and is e"ponentially distributed. achines breakdown after an average of /0 minutes operation #following a !oisson distribution$. escribe the ma&or system characteristics.
2
ANSW!S
Problem 1:
P 0 = ' −
#a$
p =
#b$
L s
20 λ = ' − = 0. => 1 0 µ
λ = 0. µ
=
#c$
20 λ = = 2 people λ − µ 0 − 20
λ 2 Lq = µ # µ − λ $ #d$
W s
=
W q
=
#e$
#f$
'
λ − µ
=
=
202 0#0 − 20$
' 0 − 2 0
λ µ # µ − λ $
=
= '. people
= 0.'0 hours
20 0#0 − 20$
= 0.0/ hours
Problem 2:
%rom the solution to !roblem ': )he average person waits 0.0/ hours and there are '0#20 arrivals 3 * hours$ arrivals per day. )herefore: )otal waiting time 4 '0 " 0.0/ 4 '0./ hours )otal cost for waiting 4 )otal waiting time 3 5ost per hour 4 '0./ 3 ('2 4 ('2* per day. -alary cost 4 * hours 3 ('0 4 (*0 )otal cost 4 -alary cost 6 7aiting cost 4 (*0 6 ('2* 4 (20* per day.
Problem 3:
= 20 per hour µ = 0 per hour M = 2 open channels #servers$
λ
#a$ P 0
=
' ' 20 ÷ 08 0
0
' 2 2#0$ ' 20 ' 20 + ÷ + ÷ '8 0 2 0 2#0$ − 20÷
=
' 2 ' 9 0 ' + + ÷ ÷ 2 : 0 − 20
=
' 2 ' '+ +
'
= = ;01 2
9
#b$ 2 20 #20$#0$ ÷ ÷ ' 20 0 ÷ + L s = ÷ 0 #'$<#2$#0$ − 20=2÷ 2 ÷
*00 ÷ = '00 =
' '2
+
* '2
' 2
+
2
= 0./; people
#c$
W s
=
L s
λ
=
0./; = 0.0/; hours 20
Problem 4:
Lq
W q
= =
λ 2 2 µ # µ − λ $
= '.'2;
λ 2 µ #µ − λ $
= 0./;
people in the ueue on average
minutes in the ueue waiting
λ = '.*/ µ people in the system
L s
= Lq +
W s
= W q +
' = 0.2; µ minutes in the system
;
Problem 5:
N 4 ; T 4 '0 minutes U 4 /0 minutes M 4 ' server X =
T T
+ U
=
'0 = 0.'2; '0 + /0
%rom )able .*: where X 4 .'2; and number of service channels 4 ', D 4 0.9/, F 4 0.20 Average number waiting 4 L 4 N #' > F $ 4 ;#' > 0.20$ 4 0.9 Average number of machines running 4 J 4 NF #' > X $ 4 ;#0.20$#' > 0.'2;$ 4 9.02; machines Average number of machines being serviced 4 H 4 FNX 4 #0.20$#;$#0.'2;$ 4 0.;/; machines !robability of no wait 4 ' > D 4 ' > 0.9/ 4 0.;2/
"a#a$it% Planning Problem 1:
)he design capacity for engine repair in our company is *0 trucks?day. )he effective capacity is 90 engines?day and the actual output is engines?day. 5alculate the utili@ation and efficiency of the operation. If the efficiency for ne"t month is e"pected to be *21, what is the e"pected output
Problem 2:
iven: F = fi"ed cost = ('000 V = variable cost = (2 ? unit P = selling price = (9? unit
%ind the break+even point in ( and in units.
Problem 3:
evelop the break+even chart for !roblem 2.
Problem 4:
CackDs rocery is manufacturing a Estore brandF item that has a variable cost of (0./; per unit and a selling price of ('.2; per unit. %i"ed costs are ('2,000. 5urrent volume is ;0,000 units. )he rocery can substantially improve the product uality by adding a new piece of euipment at an additional fi"ed cost of (;,000. Gariable cost would increase to ('.00, but their volume should increase to /0,000 units due to the higher uality product. -hould the company buy the new euipment
Problem 5:
7hat are the break+even points #( and units$ for the two processes considered in !roblem 9
Problem &:
evelop a break+even chart for !roblem 9.
Problem ':
ood Hews8 ou are going to receive (,000 in each of the ne"t ; years for sale of used machinery. A bank is willing to lend you the present value of the money in the meantime at discount of '01 per year. Jow much cash do you receive now
/
ANSW!S:
Problem 1:
Ktili@ation 4
Lfficiency 4
Actual output esign capacity
Actual output Lffective capacity
L"pected Mutput
=
*0
=
= 9;1
90
= :01
= #Lffective capacity$ #Lfficiency$ = #90$#0.*2$ = 2.* engines?day
Problem 2:
Break+even point#($ = BEP #($ =
F V
'+ Break+even point# x $ = BEP # x $ =
=
P
F P+V
'000 '000 = = (2, 000 2 0.; '+ 9
=
'000 = ;00 9−2
Problem 3:
*
Problem 4:
!rofit 4 TR – TC Mption A: -tay as is: !rofit = ;0,0003#'.2;
−
./;$ − '2,000 = (',000.
Mption B: Add euipment: !rofit = /0, 000 3 #'.2; − '.00$ − '/, 000 = (;00.
)herefore the company should continue as is with the present euipment as this returns a higher profit..
Problem 5:
Ksing current euipment: BEP #($ =
BEP# x $ =
F V
'−
=
P
F
'2, 000 0./; '− '.2;
=
P − V
=
'2, 000 '2, 000 = = (0, 000 ' − 0.0 0.90
'2,000 '.2; − 0./;
= 29, 000
Ksing the new euipment BEP #($ =
BL! # "$
F V
'−
=
P
% !
G
'/, 000 '.00 '− '.2;
=
'/, 000 '/, 000 = = (*;, 000. ' − .*0 0.2
'/,000
'/,000
'2; . '00 .
0.2;
*,000.
Problem &:
Problem ':
)he net present value factor for '01 and ; years is ./ #./ 0.0 0.*2 0./;' 0.* 0. 2'$ )herefore, the present value is: ./: 3 (,000
(22,/90
)he Bad Hews is you do have to pay back the loans8
'0
(uman !esour$es and )ob *esign Problem 1:
evelop a Process Chart for making a grilled cheese sandwich.
Problem 2:
evelop an Act!t" Chart for doing three loads of laundry.
Problem 3:
evelop a Process Chart for changing the oil in an automobile.
Problem 4:
evelop an Act!t" Chart for writing a term paper.
''
ANSW!S
Problem 1:
Mne possible solution. )he level of detail in process charts depends u pon the reuirements of the &ob. )ime is often included to aid analysis of value added. !rocess 5hart istance
-ymbol
!rocess escription
'0
ove to cabinet
+
et loaf of bread
ove to counter
+
Mpen loaf of bread
+
Nemove two slices of bread
+
Oay slices on counter+top
+
5lose loaf of bread ove to cabinet Neplace loaf of bread on shelf
'0
ove to refrigerator et mustard, package of ham from refrigerator, and butter
'0
ove to counter
+
Mpen package of ham
+
Nemove two slices of ham
+
5lose package of ham
+
Mpen mustard
+
-pread mustard on bread
+
5lose mustard
'2
+
!lace ham on bread
+
5lose sandwich
+
Mpen butter
+
-pread butter on top slice of bread
;
ove to stove
+
et fry pan
+
)urn heat on under fry pan
+
7ait for pan to heat
;
ove to counter
+
et sandwich P butter
;
ove to stove
+
!lace sandwich, buttered+side down in pan
+
Butter top slice
+
5lose butter
;
ove to counter
+
!ick up ham, mustard, and butter
'0
ove to refrigerator
+
Neturn butter, mustard, and ham to refrigerator
;
ove to stove
+
7ait for sandwich to brown on bottom
+
Inspect
+
%lip sandwich
'
+
7ait for sandwich to brown on bottom
+
Inspect sandwich
'0
ove to serving area
+
-erve sandwich
'9
Problem 2: )ime
Mperator
achine '
achine 2
7asher
ryer
Ooad Being clothes and loaded detergent in to achine '
Idle
Idle
Nun
Idle
Nemove clothes from achine '
Being unloaded
Idle
Ooad clothes into achine 2
Idle
Being loaded
Ooad Being clothes and loaded detergent into achine '
Nun
Idle
Nun
Nun
Nemove clothes from achine 2
Idle
Being unloaded
Jang clothes
Idle
Idle
Problem 3:
Mne solution might be: !rocess 5hart for 5hanging Mil in 5ar istance
- ymbol
+
5heck that needed filter is in stock
+
5heck that oil is in stock
';
0
ove to car et into car
+
-tart engine
+
Idle car to warm engine
+
rive car onto lift
+
-top engine Nelease hood catch
+
et out of car
'0
o to lift control Naise lift
'0
o to toolbo" et wrench et container for drained oil et rag
'0
7alk under lift
+
7ipe around oil drain plug
+
Ooosen oil drain plug
+
!osition container
+
Nemove oil drain plug
+
rain oil
+
7ipe around oil drain plug
+
Neplace oil drain plug
+
)ighten oil drain plug
20
Nemove container to disposal area
'
';
ove to lift control Oower lift 7ipe oil from wrench
;
ove to toolbo" Neturn wrench to tool chest et oil filter wrench from tool chest et container for drained oil
'0
ove to car engine area
+
Naise hood %ind oil filter
+
Ooosen oil filter
+
!osition container Nemove oil filter
20
)ake old filter and container of drained oil to disposal area
2;
ove to filter stock area et new filter
2;
ove to car engine area
+
7ipe around filter mount oil seal
+
Install new filter
+
)ighten new filter Nemove oil filler cap
90
ove to oil stock et oil from stock ove to car engine compartment
'/
Mpen oil containerQ pour in oil filler Neplace oil filler cap 5lean hands -tart engine Idle engine -top engine 5heck oil level 5heck oil filter seal 5heck oil drain plug 7ipe up any spilled oil )ake empty oil containers to disposal area 7ipe oil from oil filter wrench
2;
Neturn oil filter wrench to tool chest -tart engine rive car off lift !ark car for owner pickup Neturn keys
'*
Problem 4:
Activity 5hart for 7riting )erm !aper )ime
Mperator
5omputer 5omputer ' 2 esktop
evelop topic
Ksed for word processing
evelop initial outline
Ksed for word processing
Nesearch
%lesh out outline with information from research
Oibrary
Ksed for look+up and web search Ksed for word processing
Lvaluate paper %inal edit paper Ksed for word processing !roof read paper
Ksed for word processing
!rint final copy Ksed for of paper printing
oes this Activity 5hart contain enough detail that you could estimate the time it would take to write the term paper
Aggregate Planning Problem 1:
-et the following problem up in transportation format and solve for the minimum cost plan.
'
!eriod
%eb
ar
Apr
emand
;;
/0
/;
Negular
;0
;0
;0
Mvertime
;
;
;
-ubcontract
'2
'2
'0
Beginning Inventory
'0
5apacity
5osts Negular time
(0 per unit
Mvertime
(*0 per unit
-ubcontract
(0 per unit
Inventory carrying cost
(' per unit per month
Back order cost
( per unit per month
ANSW!S
Problem 1:
20
Lo$ation Strategies Problem 1:
2'
A ma&or drug store chain wishes to build a new warehouse to serve the whole idwest. At the moment, it is looking at three possible locations. )he factors, weights, and ratings being considered are given below:
Natings %actor
7eights
!eoria
es oines
5hicago
Hearness to markets
20
9
/
;
Oabor cost
;
*
*
9
)a"es
';
*
/
Hearness to suppliers
'0
'0
'0
7hich city should they choose Problem 2:
BalfourDs is considering building a plant in one of three possible locations. )hey have estimated the following parameters for each location:
Oocation
%i"ed 5ost
Gariable 5ost
7aco, )e"as
(00,000
(;./;
)i&uana, e"ico
(*00,000
(2./;
%ayetteville, Arkansas
('00,000
(*.00
%or what unit sales volume should they choose each location Problem 3:
Mur main distribution center in !hoeni", AR is due to be replaced with a much larger, more modern facility that can handle the tremendous needs that have developed with the cityDs growth. %resh produce travels to the seven store locations several times a day making site selection critical for efficient distribution. Ksing the data in the following table, determine the map coordinates for the proposed new distribution center.
22
-tore Oocations
ap 5oordinates #",y$
)ruck Nound )rips per ay
esa
#'0,;$
lendale
#,*$
5amelback
#9,/$
2
-cottsdale
#';,'0$
Apache Cunction
#',$
;
-un 5ity
#','2$
!ima
#;,;$
'0
2
Problem 4:
A company is planning on e"panding and building a new plant in one of three countries in iddle or Lastern Lurope. )he general manager, !atricia onegal, has decided to base her decision on si" critical success factors: technology availability and support, availability and uality of public education, legal and regulatory aspects, social and cultural aspects, economic factors, and political stability. Ksing a rating system of ' #least desirable$ to ; #most desirable$ she has arrived at the following ratings #you may, of course, have different opinions$. In which country should the plant be bu ilt
5ritical -uccess %actor
)urkey
-erbia
-lovakia
)echnology availability and support
9
9
Availability and uality of public education
9
9
Oegal and regulatory aspects
2
9
;
-ocial and cultural aspects
;
9
Lconomic factors
9
!olitical stability
9
2
Problem 5:
Assume that !atricia decides to use the following weights for the critical success factors: )echnology availability and support
0.
Availability and uality of public education
0.2
Oegal and regulatory aspects
0.'
-ocial and cultural aspects
0.'
Lconomic factors
0.'
!olitical stability
0.2
7ould this change her decision
29
Problem &:
!atriciaDs advisors have suggested that )urkey and -lovakia might be better differentiated by either #a$ doubling the number of critical success factors, or #b$ breaking down each of the e"isting critical success factors into smaller, more narrowly defined items, e.g., Availability and uality of public education might be broken into primary, secondary, and post+secondary education. Jow would you advise s. onegal
2;
ANSW!S: Problem 1:
Natings
7eighted Natings
7eights
!eoria
es oines
5hicago
!eoria
es oines
5hicago
Hearness to markets
20
9
/
;
*0
'90
'00
Oabor cost
;
*
*
9
90
90
20
)a"es
';
*
/
'20
';
'0;
Hearness to suppliers
'0
'0
'0
'00
0
'00
-um of 7eighted ratings:
90
/;
2;
%actor
)herefore, it appears that based upon the weights and rating, es oines should be chosen.
2
Problem 2:
)ransition between 7aco and )i&uana: 00,000
+
= x = x =
#;./; x $
*00,000
+
#2./;x $
;00, 000 ',000
)ransition between 7aco and %ayetteville: 00, 000
+
#;./; x $ 200,000 **,***
= '00,000 + = 2.2; x = x
#*.00x $
2/
Problem 3:
Hew istribution 5enter should be located at:
C x
=
#'0 3$ + #3$ + #93 2$ + #';3 $ + #'3;$ + #'3$ + #;3'0$ 2;; = = /.:/ + + 2 + + ; + + '0 2
C "
=
#;3$ + #*3$ + #/ 3 2$ + #'0 3 $ + #3;$ + #'2 3$ + #;3'0$ + + 2 + + ; + + '0
2*
=
2'9 2
= .:
Problem 4 :
5ritical -uccess %actor
)urkey
-erbia
-lovaki a
)echnology availability and support
9
9
Availability and uality of public education
9
9
Oegal and regulatory aspects
2
9
;
-ocial and cultural aspects
;
9
Lconomic factors
9
!olitical stability
9
2
2
'
22
Σ
4
Based upon her ratings of the critical success factors, !atricia should choose )urkey. %rom a practical perspective, given the small difference between the scores for )urkey and -lovakia, and the sub&ectivity of the ratings themselves, !atricia would be better advised to develop additional critical success factors, more carefully weigh the individual factorsQ or, in general, to acuire more information before making her decisions.
2
Problem 5:
5ritical -uccess %actor
7gt
)urkey
)echnology availability and support
0.
9
'.2
0.
9
'.2
Availability and uality of public education
0.2
9
0.*
9
0.*
0.
Oegal and regulatory aspects
0.'
2
0.2
9
0.9
;
.;
-ocial and cultural aspects
0.'
;
0.;
0.
9
0.9
Lconomic factors
0.'
9
0.9
0.
0.
!olitical stability
0.2
9
0.*
2
0.9
0.
Σ
-erbia
4 .
-lovaki a
.'
.
Ho, in this case, use of the weighting factors does not change the recommendation. Mne might again suggest that additional information be considered in making the decision. Problem &:
#a$ oubling the number of critical success factors. )here are two issues here. %irst, from a practical perspective there are a limited number of truly EcriticalF success factors > and these should be the ones presently being considered. Any additional factors should be of secondary or tertiary importance. -econd, given the sub&ective nature of the rating process, adding additional factors would also increase the overall margin of error of the final ratings to a degree that may eliminate any gain in differentiation arising from the use of the additional factors. )he use of a ma"imum of seven to nine critical success factors is usually appropriate. #b$ iven that oneDs ability to estimate or rate an aggregate is usually better than oneDs ability to estimate or rate the individual components of the aggregate, this approach is unlikely to provide much help.
La%out Strateg%
0
Problem 1:
As in most kitchens, the baking ovens in OoriDs Sitchen in Hew Mrleans are located in one area near the cooking burners. )he refrigerators are located ne"t to each other as are the dishwashing facilities. A work area of tabletops is set aside for cutting, mi"ing, dough rolling, and assembling of final servings, although different table areas may be reserved for each of these functions. iven the following Interdepartmental Activity atri", develop an appropriate layout for OoriDs Sitchen.
Interdepartmental Activity atri"
5ooking burners #A$
5ooking Burners #A$
Nefrigerators #B$
ishwashing #5$
7ork Area #$
+
/
'
'2
+
9
*2
+
222
Nefrigerator #B$ ishwashing #5$ 7ork Area #$
+
)he present layout is:
A
B
5
with a distance of '0 feet between ad&acent areas.
'
5omputing the Ooad 3 istance measure: Ooad 3 istance
/0
A to B
/ 3 '0
A to 5
'320
*0
A to
'230
0
B to 5
93'0
90
B to
*2320
'90
5 to
2223'0
2220
)otal
*'0
evelop a preferred layout. 7hat is the sum of the loads 3 distance of your new layout
2
Problem 2:
A firm must produce 90 units?day during an *+hour workday. )asks, times, and predecessor activities are given below.
)ask
)ime #inutes$
!redecessor#s$
A
2
+
B
2
A
5
*
+
5
L
B
%
'0
, L
9
%
J
)otal
* minutes
etermine the cycle time and the appropriate number of workstations to produce the 90 units per day.
ANSW!S
Problem 1:
%rom the Activity atri", 5 and should be ne"t to each other and A should be ne"t to 5. )he other relationships are minor by comparison. Mne possible solution is:
B
A
5
with a distance of '0 feet between ad&acent areas. 5omputing the Ooad 3 istance measure:
Ooad 3 istance A to B
/ 3 '0
/0
A to 5
'3'0
'0
A to
'2320
290
B to 5
9320
*0
B to
*230
290
5 to
2223'0
2220 /000
)otal
%urther improvement is possible. )ry analy@ing the following layouts.
A
5
B
A
5
B
Problem 2:
5ycle time
=
!roduction time available Knits reuired
=
* hrs3 0 minutes?hr 90 units
9
=
9*0 90
= '2 minutes?cycle
