Tutorial 1
1. Allico Co. manufactures tables and desks. A table requires 20 board feet of wood, and a desk requires 30 board ft of wood. Wood can be
purchased for $2 a board foot, and 30,000 board ft. are available for purchase each month. It takes 3 person-hrs of labor to make an unfinished table
and 4 person-hrs of labor to make an unfinished desk. It takes Allico 2.5 more person-hrs to convert an unfinished table into a finished table, and 3.5
person-hrs to convert an unfinished desk into a finished desk. Allico has 8000 person-hrs of labor time available each month. Allico sells unfinished
tables for $75, finished tables for $125, unfinished desks for $100, and finished desks for $160. Allico believes it can sell as many units of each
product as it makes at these prices. (a)Formulate Allico's problem as a linear program so as to maximize monthly profit. Make sure you clearly
define your decision variables in words. (b) Solve this problem on a computer.
2. The Willard Toaster Company forecasts that the demand for its main toaster product during the next month 10,000 toasters. This toaster is
assembled from three major components: a housing, a coil, and a control unit. Until now Willard has manufactured all three components. However,
the 10,000 forecast units is a new high in sales volume, and it is doubtful that the firm will have sufficient production capacity to make all the
components. The company is considering contracting with a local firm to produce at least some of the components. Each component requires three
manufacturing operations, which are performed in three departments, A, B and C. The production time requirements per unit and the available time
in each department are given below.
Department
Housing(hrs)
Coil(hrs)
Control(hrs)
Dept. Time Avail.
A
.06
.04
.10
1600
B
.09
.05
.08
1600
C
.04
.06
.02
1600
The firm's accounting department has determined the unit manufacturing cost for each component. These data, along with the purchase price
quotations from the contracting firm, are as follows:
Component
Manufact Cost
Purchase Cost
Housing
$1.75
$2.05
Coil
$1.40
$1.75
Control
$2.10
$2.65
Formulate a linear program that will minimize Willard's cost during the next month while producing at least 10,000 toasters.
Chapter 8
1. PowerComp computer is designing its computer assembly line. The assembly process requires 10 separate tasks; the tasks, the average time to
perform each task, and the predecessor tasks are listed below.
Task Time (sec)
Predecessors
Positional Weight
A
46
-
B
51
A
C
27
B
D
17
B
E
21
B
F
41
D,E
G
20
C,F
H
42
G
I
37
G
J
10
H,I
PowerComp wishes to design an assembly line so as to produce at least 40 units per hour while minimizing the number of work stations. (a)
Compute the maximum cycle time. (b) Compute the theoretical minimum number of work-stations needed. (c) Using the Ranked Positional Weight
Technique (RPWT) heuristic construct an assembly line that minimizes the number of work-stations. Clearly show the positional weights for each
task, and clearly indicate which tasks should be assigned to each work station. (d) For your solution in (c) compute the actual production rate per
hour that would result. (e) Compute the efficiency (% balance) of the line designed in (c).
2. Mary Worthington is a process designer for Mesa Manufacturing. She has been told that the firm needs to produce 2100 units of a new product
per working day. Due to rest and meal breaks, there are 840 working minutes in a work day. The following table list the tasks, precedence
relationships and average task time required to produce a relay.
Task
Time (sec)
Must follow task
A
19
-
B
6
A
C
15
B
D
15
-
E
9
D
F
18
E
G
8
E
H
9
F,G
I
11
H
J
8
H
K
6
I,J
L
22
C,K
(a) Compute the maximum cycle time Mesa can have and still produce 2100 units per day. (b) Use the Ranked Positional Weight Technique
heuristic to construct a production line that attempts to minimize the number of work-stations while producing at least 2100 units/day. Clearly show
your method and the final assignment of tasks to work stations. (c) For your solution in (b) compute the actual cycle time and the actual production
rate that would result. (d) Compute the efficiency (% balance) of the line designed in (b).
3. Elizabeth Layton is a process designer for Kennedy Manufacturing. She has been told that the firm needs to produce 2050 units of a new product
per working day. Due to rest and meal breaks, there are 840 working minutes in a work day. The following table list the tasks, precedence
relationships and average task times required to produce each unit of product.
Task
Time (sec)
Must follow task
A
6
-
B
19
A
C
14
B
D
15
-
E
9
D
F
18
E
G
8
E
H
9
F,G
I
11
H
J
9
H
K
6
I,J
L
22
C,K
(a) Compute the maximum cycle time that Kennedy Mfg. can have and still produce 2050 units per day. (b) Use the Ranked Positional Weight
Technique heuristic to construct a production line that attempts to minimize the number of work-stations while producing at least 2050 units/day.
Clearly show your method and the final assignment of tasks to work stations. (c) For your solution in (b) compute the actual cycle time and the
actual production rate that would result. (d) Compute the efficiency (% balance) of the line designed in (b).
4. A company makes a variety of products that have been divided into 8 families. To make the products the company uses six different operations
(A-F). The routing of each product family is given below along with the percentage of the company's total production volume attributed to each
family.
Family
Routing
% of Comp. volume
1
A->C->D->F
10
2
B->C->A->E
12
3
C->D->E->F->A
15
4
B->A->B->E->F
10
5
A->D->E->B->A
8
6
A->C->B->C->F->E
15
7
B->A->D->E->F->C
7
8
C->A->B->F->E->B
23
(a) Suppose the company is considering creating a flow cell to use in its operations. The company has narrowed its choices to either an A->B or a
D->E cell. Identify which flow cell would be better and justify your answer. (b) Suppose the company is considering creating a group cell to use in
its operations. The company has narrowed its choices to either an A-B-C or a B-E-F cell. Identify which group cell would be better and justify your
answer.
Chapter 9
1. Servotech Corp. has a large manufacturing plant that contains several hundred pieces of equipment. When equipment breaks down the
maintenance department is called and a maintenance worker is sent to fix the problem. Breakdowns occur randomly according to a poisson process
(the number of pieces of equipment is large enough so that the "customer" population can be treated as infinite) at the rate of 1.23 breakdowns per
hour. The time required to repair a breakdown is exponentially distributed with an average repair time of 2 hours. Servotech currently has three
maintenance workers on staff at all times. (a) Compute the average number of repair calls that are waiting in the queue (i.e., waiting for a
maintenance worker to respond to the call). (b) Compute the average time a broken machine waits until repair begins. (c) Suppose that each hour a
broken machine is out of service the company loses $50 in lost time and production. Compute the average hourly savings in lost time and
production if Servotech added a fourth maintenance worker. (d) Suppose each maintenance worker costs $30 per hour. Should Servotech add a
fourth maintenance worker? Explain why or why not.
2. Suppose customers arrive at a bank according to a Poisson process at the rate of 20 customers per hour. One teller is assigned currently to the
counter to serve the customers. Suppose the service time is exponentially distributed with a mean service time of 2 minutes. (a) Compute the
average number of customers waiting in line. (b) Compute the average length of time a customer spends in the bank (waiting and being served). (c)
Compute the probability that there are two or more customers in the bank at the same time (i.e. at least one customer waiting for service). (d)
Suppose the bank installed a new computer system that changed the information required for transactions so that customer service times were no
longer exponential, but rather the average service time was reduced to 1.8 minutes and the standard deviation was 1 minute. For this case compute
the average number of customers waiting in line (in the queue) and the average time customers spend in the bank (waiting and being served).
3. Phone calls arrive at the University's phone registration system according to a Poisson process at the rate of 60 per hour. The calls are currently
handled by two automated servers. If a call arrives at the office and both servers are busy the call is put in a queue until one of the servers is
available. Very little balking or reneging occurs, and the queue has essentially an infinite capacity. The time to service calls is approximately
exponentially distributed, with an average time of 72 seconds. (a) Compute the average number of calls waiting to be processed by a server.(b)
Compute the average time a customer spends waiting and being served. (c) Compute the probability a call has to wait in the queue. (d) Suppose the
University is considering replacing the two current servers with a single high speed automated server that could process phone calls twice as fast
(i.e. service times would be exponentially distributed, but with an average time of 36 seconds). With this single high speed server what would be the
average time a customer spends in the system (waiting and being served)?
4. Ships arrive at the Baytown dock according to a poisson process. On average a ship arrives approximately every two hours. Only one ship can be
loaded or unloaded at a time, and the time to load or unload is normally distributed with an average time of 1.4 hours and a standard deviation of 0.8
hours. (a) Compute the average number of ships waiting in line to enter the dock. (b) Compute the average length of time a ship spends in the
system (waiting and being loaded). (c) Compute the probability that a ship has to wait in line before beginning loading or unloading. (d)* Suppose
that every hour a ship is waiting in line or being loaded or unloaded (i.e, not being used productively) the ship owners are losing $500. A new crane
could be installed on the dock, which would reduce the average loading time to 1.2 hours and the standard deviation to 0.6 hour. If a new crane
would cost $150 per hour to lease and operate, should the ship owners offer to pay for a new crane? Assuming the dock operates 16 hours per day,
how much money would be gained or lost per day by installing a new crane?
5. Customers arrive at the concession (food) stand of a baseball stadium according to a poisson process at the rate of 120 per hour. The time to serve
a customer is exponentially distributed with an average time of 48 seconds. The concession stand has two people serving customers (i.e., two server
system), and a single waiting line. (a) Compute the average number of customers waiting in the line, not being served. (b) Compute the average
amount of time customers spend waiting in line and being served. (c) Compute the probability a customer has to wait in line before being served.
(d) Suppose that by adding some new automated beverage dispensers, the average service time could be reduced to 42 seconds. Compute the
average amount of time customers spend waiting in line and being served.
6. A company has a metal grinder that is part of its production process. Items to be ground enter the grinding work-station according to a poisson
process at the rate of 16 per hour. The grinding times are random with an average time of 3 minutes per item and a standard deviation of 2 minutes.
(a) Compute the average number of items waiting in the queue to be ground. (b) Compute the average time an item spends at the work-station
waiting in line and being ground. (c) Suppose the company changes the tooling on the grinder so that the average grinding time increases to 3.1
minutes per item, but the standard deviation in grinding times decreases to 0.6 minutes. Compute the average time an item will spend at the
work-station waiting in line and being ground.
Chapter 11
1. (20 pts) Composite Corp. makes high-strength carbon composite materials. Its new A-100 composite has a target impact strength of 500 pounds
per square inch (psi) with a design tolerance of 10 psi. Experiments with the production process indicate that the process should achieve this
strength level on average, and the process standard deviation is 3 psi. Composite Corp. plans to monitor the production process using statistical
process control, whereby every 15 minutes six pieces of A-100 material will be tested for strength. (a) Construct the three-sigma upper and lower
control limits for an -chart for Composite Corp.(b) Construct the three-sigma upper and lower control limits for an R-chart for Composite Corp.(c)
Suppose the strength levels for a sample of six items were 502, 504, 508, 499, 506, 505. Is the process under control? Explain why or why not. (d)
Suppose the strength readings for a sample were 515, 522, 518, 513, 520, 508. Should the production process be stopped and adjusted so that the
strength readings are closer to 500 psi? Explain what should be done.
2. Lester Paper Company produces a specialty paper that is supposed to have a surface brightness rating of 4.50. The company uses statistical
process control to monitor its paper-making process. Every 10 minutes an automated monitoring system makes five measurements of the surface
brightness of the paper being made (i.e., a sample of size 5 is taken). Below are the readings for 12 samples that were taken when the process was
believed to be under control.
Sample #
Surface Brightness
Range
1
4.52
4.55
4.51
4.49
4.54
4.522
0.06
2
4.54
4.47
4.48
4.53
4.50
4.504
0.07
3
4.45
4.46
4.51
4.47
4.56
4.488
0.11
4
4.57
4.58
4.51
4.52
4.50
4.536
0.08
5
4.46
4.51
4.49
4.53
4.48
4.494
0.07
6
4.52
4.44
4.50
4.54
4.51
4.502
0.10
7
4.50
4.49
4.58
4.52
4.49
4.516
0.09
8
4.55
4.58
4.53
4.47
4.51
4.528
0.11
9
4.53
4.51
4.43
4.48
4.55
4.500
0.12
10
4.52
4.49
4.54
4.48
4.47
4.500
0.07
11
4.46
4.54
4.49
4.53
4.52
4.508
0.08
12
4.51
4.45
4.43
4.50
4.49
4.476
0.08
(a) Using , compute an estimate for the process standard deviation, sigma. (b) Compute the upper and lower control limits for a 3-sigma -chart
assuming the sample size will be 5. (c) Compute the upper and lower control limits for a 3-sigma R-chart assuming the sample size will be 5. (d)
Suppose the following surface brightness readings were taken for a sample: 4.56, 4.51, 4.44, 4.52, and 4.42. Do the control charts indicate that the
process is in-control or out-of-control? Explain.
3. Sharp's Ammunition Company makes bullets for the military. When the production process is operating well the proportion of bullets that misfire
is 0.0005. (a) Compute the control limits for a three-sigma p-chart for Sharp's, assuming that Sharp's will test-fire 100 bullets every two hours to
evaluate whether the process is under control. (b) Suppose for a sample of 100 bullets, one bullet misfires. Explain what this means and what
Sharp's should do.
Chapter 13
1. A pharmacy sells an average of 50 VHS (video cassette) tapes per day. The supplier charges the pharmacy $1.50 per tape, and there is a fixed
delivery charge of $90 for any purchase of 10,000 tapes or less (the number that can fit in the delivery truck). The pharmacy believes that its holding
cost rate on that product is 24% per year of the item's value. (Assume the pharmacy is open 360 days/year.)
(a) Compute the optimal number of tapes the pharmacy should purchase at a time. (b) Suppose the supplier offers to charge only $1.48 per tape if
the pharmacy buys at least 5000 tapes at a time, and a price of only $1.46 if the pharmacy buys 8000 at a time. Compute the pharmacy's optimal
order quantity. (c) Suppose the demand during lead-time (DDLT) is normally distributed with a mean value of 350 and a standard deviation of 100.
Suppose the pharmacy uses a reorder point of 400 (i.e., it places an order when the inventory position hits 400). Using your order quantity from part
(a), compute the demand service level (SLd) that results from using this reorder point.
2. Atlas Corp. uses 20,000 pounds of chemical dye a year. The annual holding cost is 20% per year of the purchase price, and it costs Atlas $320 to
place an order and receive delivery. The supplier offers Atlas a quantity discount on the dye as follows:
Quantity
Bought
Price
per Pound
1 to 999
$4.20
1000 to 4999
$4.00
5000 or more
$3.80
(a) Determine the optimal amount of dye Atlas should buy each time it orders. (b) Compute the total material costs per year for Atlas if it uses the
order quantity from (a).
3. John Wiley and Sons must determine how many copies of a new Taxation textbook to print. Wiley will make only one print run because it
revises the book annually and the printing schedule makes a second run later in the year impossible. Each copy of the book costs $12 to print and
bind (variable cost). Wiley sells the book to distributors for $40 a copy, and any unsold book at the end of the year is scrapped (with a value of 0). The demand for
the book during the next year is random, but Wiley's marketing department has developed the following probability distribution for demand.
Demand
Probability
10,000
0.10
15,000
0.15
20,000
0.20
22,000
0.20
24,000
0.10
26,000
0.10
28,000
0.10
30,000
0.05
(a) Determine the optimal number of copies of the book that Wiley should print to maximize expected profit. (b) Compute the expected (average)
profit that would result from your answer in part (a).
4. Suppose a company uses an average of 1000 units per day of a component, and the demand during lead-time (DDLT) is normally distributed
with a mean of 5000 and a standard deviation of 1500. What should the company's reorder point be (i.e., what should its inventory position be when
it places an order), so as to achieve a 99% demand service level (SLd)? Assume the order quantity is 25,000 units each cycle.
5. Devon Corp. uses 40,000 pounds of chemical dye a year. The annual holding cost is 20% per year of the purchase price, the purchase price is
$4.20 per pound, and it costs Devon $640 to place an order and receive delivery (regardless of order quantity). (a) Determine the optimal amount of
dye Devon should buy each time it orders. (b) How many orders will Devon place each year, using the order quantity derived in part (a)? (c)
Compute the total annual stocking cost of the policy derived in part (a). (d) Suppose the supplier of the dye offered to sell the dye to Devon for
$4.10 per pound, if Devon would buy 20,000 pounds per order. Should Devon accept this offer? Justify your answer.
6. A pharmacy sells an average of 50 VHS (video cassette) tapes per day. The supplier charges the pharmacy $1.50 per tape, and there is a fixed
delivery charge of $90 for any purchase of 10,000 tapes or less (the number that can fit in the delivery truck). The pharmacy believes that its holding
cost rate on that product is 24% per year of the item's value. (Assume the pharmacy is open 360 days/year.) (a) Compute the optimal number of
tapes the pharmacy should purchase at a time. (b) Suppose the supplier offers to charge only $1.48 per tape if the pharmacy buys at least 5000 tapes
at a time, and a price of only $1.46 if the pharmacy buys 8000 at a time. Compute the pharmacy's optimal order quantity. (c) Suppose the demand
during lead-time (DDLT) is normally distributed with a mean value of 350 and a standard deviation of 100. Suppose the pharmacy uses a reorder
point of 400 (i.e., it places an order when the inventory position hits 400). USING YOUR ORDER QUANTITY FROM PART (a), compute the
demand service level (SLd) that results from using this reorder point.
Chapter 15
1. A company makes three products, A, B, and C, using a three-stage flow process. The company uses a Just-In-Time (kanban) production system.
Because of differences in demand the company produces product A in lots of 100 units, B in lots of 150 units, and C in lots of 200 units. At each of
the three stages changing from one product to another requires a 1-hour set-up, and the company can process 50 units per hour at each stage when
the stage is operating. Below is a table showing the operation of the factory during the past 8 hours. The table also lists the actual demands for the
products during the next five hours.
Fill in the table for the next 5 hours indicating what is being done at each production stage, where and when kanbans are issued, how materials flow
between stages, and what the inventory levels are at each stage and for the final products. Assume that materials completed at one stage are
available at the next stage in the next hour (as shown on the figure below). Raw materials requested are received from the supplier one hour later.
Day Raw Stage 1 Stage 2 Stage 3 Prod. Inv. Demands
Hour Mat'l Actv. Kanb Inv. Actv Kanb Inv Actv Kanb A B C A B C
1 BAC run C BA ABC SU C BA BC run A CB 180 240 270 15 30 10
2 BAC run C BAC AB run C BA BC run A CB 165 210 260 20 25 20
3 BAC run C BAC AB run C BA BC SU C B 245 185 240 10 5 25
4 BAC SU B AC CAB run C BAC B run C B 235 180 215 15 25 40
5 AC run B AC CAB run C BAC B run C B 220 155 175 5 10 20
6 BAC run B AC CAB SU B AC CB run C BC 215 145 155 10 15 25
7 BAC run B ACB CA run B AC CB run C BCB 205 130 130 10 10 15
8 BAC SU A CB BCA run B AC CB SU B CB 195 120 315 5 15 20
9 ___ ____ ___ ___ _____ ___ ___ ____ ___ ____ ___ ___ 10 20 25
10 ___ ____ ___ ___ _____ ___ ___ ____ ___ ____ ___ ___ 5 10 15
11 ___ ____ ___ ___ _____ ___ ___ ____ ___ ____ ___ ___ 10 15 20
12 ___ ____ ___ ___ _____ ___ ___ ____ ___ ____ ___ ___ 5 10 15
13 ___ ____ ___ ___ _____ ___ ___ ____ ___ ____ ___ ___ 10 15 15
Chapter 16
1. An auto repair shop has five mechanics that must be assigned to repair five waiting cars. From the types of repairs needed the supervisor knows
the time each mechanic needs to do the job. These times, in hours, are shown below.
Mechanic
Job
1
2
3
4
5
Alice
3
5
2
4
6
Bob
4
6
4
3
7
Carol
2
7
3
4
5
Darren
5
4
7
3
6
Ed
3
4
4
5
5
Assuming each mechanic can be assigned to only 1 car, determine the minimum cost assignment using the Hungarian algorithm. Clearly identify
the best assignments (people to jobs).
Wiley's College Division
URL:http://www.umsl.edu/~jmartini/book/additionalproblems.html
Page Owner: Joseph Martinich (Joseph.Martinich@umsl.edu)
Last Updated: May 22, 1998
Tutorial 1
1. Allico Co. manufactures tables and desks. A table requires 20 board feet of wood, and a desk requires 30 board ft of wood. Wood can be purchased for $2 a board foot, and 30,000 board ft. are available for purchase each month. It takes 3 person-hrs of labor to make an unfinished table and 4 person-hrs of labor to make an unfinished desk. It takes Allico 2.5 more person-hrs to convert an unfinished table into a finished table, and 3.5 person-hrs to convert an unfinished desk into a finished desk. Allico has 8000 person-hrs of labor time available each month. Allico sells unfinished tables for $75, finished tables for $125, unfinished desks for $100, and finished desks for $160. Allico believes it can sell as many units of each product as it makes at these prices. (a)Formulate Allico's problem as a linear program so as to maximize monthly profit. Make sure you clearly define your decision variables in words. (b) Solve this problem on a computer. 2. The Willard Toaster Company forecasts that the demand for its main toaster product during the next month 10,000 toasters. This toaster is assembled from three major components: a housing, a coil, and a control unit. Until now Willard has manufactured all three components. However, the 10,000 forecast units is a new high in sales volume, and it is doubtful that the firm will have sufficient production capacity to make all the components. The company is considering contracting with a local firm to produce at least some of the components. Each component requires three manufacturing operations, which are performed in three departments, A, B and C. The production time requirements per unit and the available time in each department are given below. |
Department |
Housing(hrs) |
Coil(hrs) |
Control(hrs) |
Dept. Time Avail. |
A |
.06 |
.04 |
.10 |
1600 |
B |
.09 |
.05 |
.08 |
1600 |
C |
.04 |
.06 |
.02 |
1600 |
The firm's accounting department has determined the unit manufacturing cost for each component. These data, along with the purchase price quotations from the contracting firm, are as follows: |
Component |
Manufact Cost |
Purchase Cost |
Housing |
$1.75 |
$2.05 |
Coil |
$1.40 |
$1.75 |
Control |
$2.10 |
$2.65 |
Formulate a linear program that will minimize Willard's cost during the next month while producing at least 10,000 toasters. |
Chapter 8
1. PowerComp computer is designing its computer assembly line. The assembly process requires 10 separate tasks; the tasks, the average time to perform each task, and the predecessor tasks are listed below. |
Task Time (sec) |
Predecessors |
Positional Weight |
A |
46 |
- |
B |
51 |
A |
C |
27 |
B |
D |
17 |
B |
E |
21 |
B |
F |
41 |
D,E |
G |
20 |
C,F |
H |
42 |
G |
I |
37 |
G |
J |
10 |
H,I |
PowerComp wishes to design an assembly line so as to produce at least 40 units per hour while minimizing the number of work stations. (a) Compute the maximum cycle time. (b) Compute the theoretical minimum number of work-stations needed. (c) Using the Ranked Positional Weight Technique (RPWT) heuristic construct an assembly line that minimizes the number of work-stations. Clearly show the positional weights for each task, and clearly indicate which tasks should be assigned to each work station. (d) For your solution in (c) compute the actual production rate per hour that would result. (e) Compute the efficiency (% balance) of the line designed in (c). 2. Mary Worthington is a process designer for Mesa Manufacturing. She has been told that the firm needs to produce 2100 units of a new product per working day. Due to rest and meal breaks, there are 840 working minutes in a work day. The following table list the tasks, precedence relationships and average task time required to produce a relay. |
Task |
Time (sec) |
Must follow task |
A |
19 |
- |
B |
6 |
A |
C |
15 |
B |
D |
15 |
- |
E |
9 |
D |
F |
18 |
E |
G |
8 |
E |
H |
9 |
F,G |
I |
11 |
H |
J |
8 |
H |
K |
6 |
I,J |
L |
22 |
C,K |
(a) Compute the maximum cycle time Mesa can have and still produce 2100 units per day. (b) Use the Ranked Positional Weight Technique heuristic to construct a production line that attempts to minimize the number of work-stations while producing at least 2100 units/day. Clearly show your method and the final assignment of tasks to work stations. (c) For your solution in (b) compute the actual cycle time and the actual production rate that would result. (d) Compute the efficiency (% balance) of the line designed in (b). 3. Elizabeth Layton is a process designer for Kennedy Manufacturing. She has been told that the firm needs to produce 2050 units of a new product per working day. Due to rest and meal breaks, there are 840 working minutes in a work day. The following table list the tasks, precedence relationships and average task times required to produce each unit of product. |
Task |
Time (sec) |
Must follow task |
A |
6 |
- |
B |
19 |
A |
C |
14 |
B |
D |
15 |
- |
E |
9 |
D |
F |
18 |
E |
G |
8 |
E |
H |
9 |
F,G |
I |
11 |
H |
J |
9 |
H |
K |
6 |
I,J |
L |
22 |
C,K |
(a) Compute the maximum cycle time that Kennedy Mfg. can have and still produce 2050 units per day. (b) Use the Ranked Positional Weight Technique heuristic to construct a production line that attempts to minimize the number of work-stations while producing at least 2050 units/day. Clearly show your method and the final assignment of tasks to work stations. (c) For your solution in (b) compute the actual cycle time and the actual production rate that would result. (d) Compute the efficiency (% balance) of the line designed in (b). 4. A company makes a variety of products that have been divided into 8 families. To make the products the company uses six different operations (A-F). The routing of each product family is given below along with the percentage of the company's total production volume attributed to each family. |
Family |
Routing |
% of Comp. volume |
1 |
A->C->D->F |
10 |
2 |
B->C->A->E |
12 |
3 |
C->D->E->F->A |
15 |
4 |
B->A->B->E->F |
10 |
5 |
A->D->E->B->A |
8 |
6 |
A->C->B->C->F->E |
15 |
7 |
B->A->D->E->F->C |
7 |
8 |
C->A->B->F->E->B |
23 |
(a) Suppose the company is considering creating a flow cell to use in its operations. The company has narrowed its choices to either an A->B or a D->E cell. Identify which flow cell would be better and justify your answer. (b) Suppose the company is considering creating a group cell to use in its operations. The company has narrowed its choices to either an A-B-C or a B-E-F cell. Identify which group cell would be better and justify your answer. |
Chapter 9
1. Servotech Corp. has a large manufacturing plant that contains several hundred pieces of equipment. When equipment breaks down the maintenance department is called and a maintenance worker is sent to fix the problem. Breakdowns occur randomly according to a poisson process (the number of pieces of equipment is large enough so that the "customer" population can be treated as infinite) at the rate of 1.23 breakdowns per hour. The time required to repair a breakdown is exponentially distributed with an average repair time of 2 hours. Servotech currently has three maintenance workers on staff at all times. (a) Compute the average number of repair calls that are waiting in the queue (i.e., waiting for a maintenance worker to respond to the call). (b) Compute the average time a broken machine waits until repair begins. (c) Suppose that each hour a broken machine is out of service the company loses $50 in lost time and production. Compute the average hourly savings in lost time and production if Servotech added a fourth maintenance worker. (d) Suppose each maintenance worker costs $30 per hour. Should Servotech add a fourth maintenance worker? Explain why or why not. 2. Suppose customers arrive at a bank according to a Poisson process at the rate of 20 customers per hour. One teller is assigned currently to the counter to serve the customers. Suppose the service time is exponentially distributed with a mean service time of 2 minutes. (a) Compute the average number of customers waiting in line. (b) Compute the average length of time a customer spends in the bank (waiting and being served). (c) Compute the probability that there are two or more customers in the bank at the same time (i.e. at least one customer waiting for service). (d) Suppose the bank installed a new computer system that changed the information required for transactions so that customer service times were no longer exponential, but rather the average service time was reduced to 1.8 minutes and the standard deviation was 1 minute. For this case compute the average number of customers waiting in line (in the queue) and the average time customers spend in the bank (waiting and being served). 3. Phone calls arrive at the University's phone registration system according to a Poisson process at the rate of 60 per hour. The calls are currently handled by two automated servers. If a call arrives at the office and both servers are busy the call is put in a queue until one of the servers is available. Very little balking or reneging occurs, and the queue has essentially an infinite capacity. The time to service calls is approximately exponentially distributed, with an average time of 72 seconds. (a) Compute the average number of calls waiting to be processed by a server.(b) Compute the average time a customer spends waiting and being served. (c) Compute the probability a call has to wait in the queue. (d) Suppose the University is considering replacing the two current servers with a single high speed automated server that could process phone calls twice as fast (i.e. service times would be exponentially distributed, but with an average time of 36 seconds). With this single high speed server what would be the average time a customer spends in the system (waiting and being served)? 4. Ships arrive at the Baytown dock according to a poisson process. On average a ship arrives approximately every two hours. Only one ship can be loaded or unloaded at a time, and the time to load or unload is normally distributed with an average time of 1.4 hours and a standard deviation of 0.8 hours. (a) Compute the average number of ships waiting in line to enter the dock. (b) Compute the average length of time a ship spends in the system (waiting and being loaded). (c) Compute the probability that a ship has to wait in line before beginning loading or unloading. (d)* Suppose that every hour a ship is waiting in line or being loaded or unloaded (i.e, not being used productively) the ship owners are losing $500. A new crane could be installed on the dock, which would reduce the average loading time to 1.2 hours and the standard deviation to 0.6 hour. If a new crane would cost $150 per hour to lease and operate, should the ship owners offer to pay for a new crane? Assuming the dock operates 16 hours per day, how much money would be gained or lost per day by installing a new crane? 5. Customers arrive at the concession (food) stand of a baseball stadium according to a poisson process at the rate of 120 per hour. The time to serve a customer is exponentially distributed with an average time of 48 seconds. The concession stand has two people serving customers (i.e., two server system), and a single waiting line. (a) Compute the average number of customers waiting in the line, not being served. (b) Compute the average amount of time customers spend waiting in line and being served. (c) Compute the probability a customer has to wait in line before being served. (d) Suppose that by adding some new automated beverage dispensers, the average service time could be reduced to 42 seconds. Compute the average amount of time customers spend waiting in line and being served. 6. A company has a metal grinder that is part of its production process. Items to be ground enter the grinding work-station according to a poisson process at the rate of 16 per hour. The grinding times are random with an average time of 3 minutes per item and a standard deviation of 2 minutes. (a) Compute the average number of items waiting in the queue to be ground. (b) Compute the average time an item spends at the work-station waiting in line and being ground. (c) Suppose the company changes the tooling on the grinder so that the average grinding time increases to 3.1 minutes per item, but the standard deviation in grinding times decreases to 0.6 minutes. Compute the average time an item will spend at the work-station waiting in line and being ground. |
Chapter 11
1. (20 pts) Composite Corp. makes high-strength carbon composite materials. Its new A-100 composite has a target impact strength of 500 pounds per square inch (psi) with a design tolerance of 10 psi. Experiments with the production process indicate that the process should achieve this strength level on average, and the process standard deviation is 3 psi. Composite Corp. plans to monitor the production process using statistical process control, whereby every 15 minutes six pieces of A-100 material will be tested for strength. (a) Construct the three-sigma upper and lower control limits for an -chart for Composite Corp.(b) Construct the three-sigma upper and lower control limits for an R-chart for Composite Corp.(c) Suppose the strength levels for a sample of six items were 502, 504, 508, 499, 506, 505. Is the process under control? Explain why or why not. (d) Suppose the strength readings for a sample were 515, 522, 518, 513, 520, 508. Should the production process be stopped and adjusted so that the strength readings are closer to 500 psi? Explain what should be done. 2. Lester Paper Company produces a specialty paper that is supposed to have a surface brightness rating of 4.50. The company uses statistical process control to monitor its paper-making process. Every 10 minutes an automated monitoring system makes five measurements of the surface brightness of the paper being made (i.e., a sample of size 5 is taken). Below are the readings for 12 samples that were taken when the process was believed to be under control. |
Sample # |
Surface Brightness |
Range | |||||
1 |
4.52 |
4.55 |
4.51 |
4.49 |
4.54 |
4.522 |
0.06 |
2 |
4.54 |
4.47 |
4.48 |
4.53 |
4.50 |
4.504 |
0.07 |
3 |
4.45 |
4.46 |
4.51 |
4.47 |
4.56 |
4.488 |
0.11 |
4 |
4.57 |
4.58 |
4.51 |
4.52 |
4.50 |
4.536 |
0.08 |
5 |
4.46 |
4.51 |
4.49 |
4.53 |
4.48 |
4.494 |
0.07 |
6 |
4.52 |
4.44 |
4.50 |
4.54 |
4.51 |
4.502 |
0.10 |
7 |
4.50 |
4.49 |
4.58 |
4.52 |
4.49 |
4.516 |
0.09 |
8 |
4.55 |
4.58 |
4.53 |
4.47 |
4.51 |
4.528 |
0.11 |
9 |
4.53 |
4.51 |
4.43 |
4.48 |
4.55 |
4.500 |
0.12 |
10 |
4.52 |
4.49 |
4.54 |
4.48 |
4.47 |
4.500 |
0.07 |
11 |
4.46 |
4.54 |
4.49 |
4.53 |
4.52 |
4.508 |
0.08 |
12 |
4.51 |
4.45 |
4.43 |
4.50 |
4.49 |
4.476 |
0.08 |
(a) Using , compute an estimate for the process standard deviation, sigma. (b) Compute the upper and lower control limits for a 3-sigma -chart assuming the sample size will be 5. (c) Compute the upper and lower control limits for a 3-sigma R-chart assuming the sample size will be 5. (d) Suppose the following surface brightness readings were taken for a sample: 4.56, 4.51, 4.44, 4.52, and 4.42. Do the control charts indicate that the process is in-control or out-of-control? Explain. 3. Sharp's Ammunition Company makes bullets for the military. When the production process is operating well the proportion of bullets that misfire is 0.0005. (a) Compute the control limits for a three-sigma p-chart for Sharp's, assuming that Sharp's will test-fire 100 bullets every two hours to evaluate whether the process is under control. (b) Suppose for a sample of 100 bullets, one bullet misfires. Explain what this means and what Sharp's should do. |
Chapter 13
1. A pharmacy sells an average of 50 VHS (video cassette) tapes per day. The supplier charges the pharmacy $1.50 per tape, and there is a fixed delivery charge of $90 for any purchase of 10,000 tapes or less (the number that can fit in the delivery truck). The pharmacy believes that its holding cost rate on that product is 24% per year of the item's value. (Assume the pharmacy is open 360 days/year.) (a) Compute the optimal number of tapes the pharmacy should purchase at a time. (b) Suppose the supplier offers to charge only $1.48 per tape if the pharmacy buys at least 5000 tapes at a time, and a price of only $1.46 if the pharmacy buys 8000 at a time. Compute the pharmacy's optimal order quantity. (c) Suppose the demand during lead-time (DDLT) is normally distributed with a mean value of 350 and a standard deviation of 100. Suppose the pharmacy uses a reorder point of 400 (i.e., it places an order when the inventory position hits 400). Using your order quantity from part (a), compute the demand service level (SLd) that results from using this reorder point. 2. Atlas Corp. uses 20,000 pounds of chemical dye a year. The annual holding cost is 20% per year of the purchase price, and it costs Atlas $320 to place an order and receive delivery. The supplier offers Atlas a quantity discount on the dye as follows: |
Bought | Price |
1 to 999 | $4.20 |
1000 to 4999 | $4.00 |
5000 or more | $3.80 |
(a) Determine the optimal amount of dye Atlas should buy each time it orders. (b) Compute the total material costs per year for Atlas if it uses the order quantity from (a). 3. John Wiley and Sons must determine how many copies of a new Taxation textbook to print. Wiley will make only one print run because it revises the book annually and the printing schedule makes a second run later in the year impossible. Each copy of the book costs $12 to print and bind (variable cost). Wiley sells the book to distributors for $40 a copy, and any unsold book at the end of the year is scrapped (with a value of 0). The demand for the book during the next year is random, but Wiley's marketing department has developed the following probability distribution for demand. |
|
|
10,000 | 0.10 |
15,000 | 0.15 |
20,000 | 0.20 |
22,000 | 0.20 |
24,000 | 0.10 |
26,000 | 0.10 |
28,000 | 0.10 |
30,000 | 0.05 |
(a) Determine the optimal number of copies of the book that Wiley should print to maximize expected profit. (b) Compute the expected (average) profit that would result from your answer in part (a). 4. Suppose a company uses an average of 1000 units per day of a component, and the demand during lead-time (DDLT) is normally distributed with a mean of 5000 and a standard deviation of 1500. What should the company's reorder point be (i.e., what should its inventory position be when it places an order), so as to achieve a 99% demand service level (SLd)? Assume the order quantity is 25,000 units each cycle. 5. Devon Corp. uses 40,000 pounds of chemical dye a year. The annual holding cost is 20% per year of the purchase price, the purchase price is $4.20 per pound, and it costs Devon $640 to place an order and receive delivery (regardless of order quantity). (a) Determine the optimal amount of dye Devon should buy each time it orders. (b) How many orders will Devon place each year, using the order quantity derived in part (a)? (c) Compute the total annual stocking cost of the policy derived in part (a). (d) Suppose the supplier of the dye offered to sell the dye to Devon for $4.10 per pound, if Devon would buy 20,000 pounds per order. Should Devon accept this offer? Justify your answer. 6. A pharmacy sells an average of 50 VHS (video cassette) tapes per day. The supplier charges the pharmacy $1.50 per tape, and there is a fixed delivery charge of $90 for any purchase of 10,000 tapes or less (the number that can fit in the delivery truck). The pharmacy believes that its holding cost rate on that product is 24% per year of the item's value. (Assume the pharmacy is open 360 days/year.) (a) Compute the optimal number of tapes the pharmacy should purchase at a time. (b) Suppose the supplier offers to charge only $1.48 per tape if the pharmacy buys at least 5000 tapes at a time, and a price of only $1.46 if the pharmacy buys 8000 at a time. Compute the pharmacy's optimal order quantity. (c) Suppose the demand during lead-time (DDLT) is normally distributed with a mean value of 350 and a standard deviation of 100. Suppose the pharmacy uses a reorder point of 400 (i.e., it places an order when the inventory position hits 400). USING YOUR ORDER QUANTITY FROM PART (a), compute the demand service level (SLd) that results from using this reorder point. |
Chapter 15
1. A company makes three products, A, B, and C, using a three-stage flow process. The company uses a Just-In-Time (kanban) production system. Because of differences in demand the company produces product A in lots of 100 units, B in lots of 150 units, and C in lots of 200 units. At each of the three stages changing from one product to another requires a 1-hour set-up, and the company can process 50 units per hour at each stage when the stage is operating. Below is a table showing the operation of the factory during the past 8 hours. The table also lists the actual demands for the products during the next five hours.Fill in the table for the next 5 hours indicating what is being done at each production stage, where and when kanbans are issued, how materials flow between stages, and what the inventory levels are at each stage and for the final products. Assume that materials completed at one stage are available at the next stage in the next hour (as shown on the figure below). Raw materials requested are received from the supplier one hour later. Day Raw Stage 1 Stage 2 Stage 3 Prod. Inv. DemandsHour Mat'l Actv. Kanb Inv. Actv Kanb Inv Actv Kanb A B C A B C 1 BAC run C BA ABC SU C BA BC run A CB 180 240 270 15 30 102 BAC run C BAC AB run C BA BC run A CB 165 210 260 20 25 20 3 BAC run C BAC AB run C BA BC SU C B 245 185 240 10 5 25 4 BAC SU B AC CAB run C BAC B run C B 235 180 215 15 25 40 5 AC run B AC CAB run C BAC B run C B 220 155 175 5 10 20 6 BAC run B AC CAB SU B AC CB run C BC 215 145 155 10 15 25 7 BAC run B ACB CA run B AC CB run C BCB 205 130 130 10 10 15 8 BAC SU A CB BCA run B AC CB SU B CB 195 120 315 5 15 20 9 ___ ____ ___ ___ _____ ___ ___ ____ ___ ____ ___ ___ 10 20 25 10 ___ ____ ___ ___ _____ ___ ___ ____ ___ ____ ___ ___ 5 10 15 11 ___ ____ ___ ___ _____ ___ ___ ____ ___ ____ ___ ___ 10 15 20 12 ___ ____ ___ ___ _____ ___ ___ ____ ___ ____ ___ ___ 5 10 15 13 ___ ____ ___ ___ _____ ___ ___ ____ ___ ____ ___ ___ 10 15 15 |
Chapter 16
1. An auto repair shop has five mechanics that must be assigned to repair five waiting cars. From the types of repairs needed the supervisor knows the time each mechanic needs to do the job. These times, in hours, are shown below. |
| Job | ||||
1 | 2 | 3 | 4 | 5 | |
| 3 | 5 | 2 | 4 | 6 |
| 4 | 6 | 4 | 3 | 7 |
| 2 | 7 | 3 | 4 | 5 |
| 5 | 4 | 7 | 3 | 6 |
| 3 | 4 | 4 | 5 | 5 |
Assuming each mechanic can be assigned to only 1 car, determine the minimum cost assignment using the Hungarian algorithm. Clearly identify the best assignments (people to jobs). |
Wiley's College Division
URL:http://www.umsl.edu/~jmartini/book/additionalproblems.html
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Last Updated: May 22, 1998
Copyright 1997, John Wiley & Sons, Inc.