(Solved by Humans)-Problem - 1 In May of 1998, Benjamin's Trucks, a dealership

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Problem ? 1?

In May of 1998, Benjamin?s Trucks, a dealership located in West Bloomfield, Michigan was trying to determine how many 1999 models the company should order. Each Truck cost the dealer $40,000. The demand for the dealer?s 1999 models has the probability distribution shown in Table 1 below. Each Truck is sold for $50,000. If the demand for 1999 Trucks exceeds the number of Trucks ordered in May, 1998, the dealer must order at a cost of $45,000 per Truck. If the demand for 1999 Trucks falls short, the Dealer may dispose of the excess Trucks in an end of model year sale for $35,000 per Truck.

How nay models should Benjamin?s Trucks order in May of 1998.Determine the optimal order quantityReorder PointSafety Stock LevelWhat unrealistic assumptions must be madeWhat (s, S) policy will be used in this situation.

Problem 3 ?

Assume that Michigan has only three towns (Detroit, West Bloomfield and Imlay City). Then, each Michigan Family is classified as living in Detroit (D), Imlay City (I) or West Bloomfield (W) areas. During the year 2015, 15% of all Detroit families move to West Bloomfield area and 5% move to Imlay City area. Also, 6% of all West Bloomfield families move to Detroit area and 4% move to a Imlay City area. Finally, 4% of all Imlay City families move to Detroit area and 6% move to West Bloomfield area.

• If a family now lives in Detroit area, what is the probability that the family will live in an Detroit area two years from now. ?

Hint: First set up the Probability distribution matrix ?p? ?and then use it to solve the problem.

????????????????????????????????????????? D??????????? W??????????????? I

??????????????????????? D???????????? x1,1?????? ????????x2,1??????????? x3,1??????????????????

??????????? ?p =???? W??????????? x21???????? x22?????????? x23?????? ?

???????????????????????? I????????????? x31??????? x32??????????? x33

Where;

Xij ?= percentage of families moving from city to city in Michigan

Suppose that at present, 40% of all families live in Detroit area, 35% live in West Bloomfield area, and 25% live in Imlay City Area. Two years from now, what percent of Michigan families will live in Detroit area?What problem might occur if this model was used to predict the future population distribution of the state of Michigan?

Problem 4 ?

• The time between buses follows the mass function shown in Table 2 below. What is the average length of time one must wait for a bus. Also draw the waiting time Paradox line graph

Table 2: Mass Function

Time between

Busses????????????????? ??????????????Probability

30 Minutes?????????????????? ?????????1/4

1 Hour??????????????????????????????????? 1/4

2 Hours?????????????????????????????????? ?

There are four sections of 10^th grade class at Walled Lake High School in Michigan. The number in each section is as follows Section 1, 20 students; Section 2, 25 Students; Section 3, 35 Students; and Section 4, 40 Students.What is the average size of 10^th grade section?Suppose the high school randomly selects a Walled lake 10^th grader that is a female. On the average, how many students will be in her class

Problem 5?

Each Airline passenger and his or her luggage must be checked to determine whether he or she is carrying weapons onto the airplane. Suppose that at Detroit City Airport, an average of 10 passengers per minute arrive (inter-arrival time are exponential). To check passengers for weapons the airport must have a checkpoint consisting of a metal detector and baggage X-ray machine. Whenever a checkpoint is in operation, two employees are required. A checkpoint can check an average of 12 passengers per minute (The time to check a passenger is exponential).? Under the assumption that the airport has only one checkpoint, answer the following questions.

• What is the probability that a passenger will have to wait before being checked for weapons?

• On the average, how many passengers are waiting in line to enter the checkpoint?

?

• On the average, how long will a passenger spend at a checkpoint?

Problem ? 1
In May of 1998, Benjamin?s Trucks, a dealership located in West Bloomfield, Michigan was trying to
determine how many 1999 models the company should order. Each Truck cost the dealer $40,000. The
demand for the dealer?s 1999 models has the probability distribution shown in Table 1 below. Each Truck
is sold for $50,000. If the demand for 1999 Trucks exceeds the number of Trucks ordered in May, 1998,
the dealer must order at a cost of $45,000 per Truck. If the demand for 1999 Trucks falls short, the
Dealer may dispose of the excess Trucks in an end of model year sale for $35,000 per Truck.
a. How nay models should Benjamin?s Trucks order in May of 1998.
Table 1: Probability Distribution of Trucks
# of Trucks Demanded
Probability
20
.30
25
.15
30
.15
35
.20
40
.20

Problem 2

A hospital orders its blood bank from a regional blood bank. Each year, the hospital uses an
average of 1,040 pints of type ?A? blood. For each order that the hospital places with any
regional blood bank, a cost of $20.00 was incurred. Each order?s lead time is 1 week. Inventory
cost per pint of blood per year is $20.00. The per pint stock-out cost is estimated to be $50.00.
Annual demand for Type ?A? blood is normally distributed with standard deviation of 43.26 pints.
a. Determine the optimal order quantity
b. Reorder Point
c. Safety Stock Level
Assume that 52 weeks = 1 year and that all demand is backlogged. To use the techniques of this section,
d. What unrealistic assumptions must be made
e. What (s, S) policy will be used in this situation.
Problem 3

Assume that Michigan has only three towns (Detroit, West Bloomfield and Imlay City). Then,
each Michigan Family is classified as living in Detroit (D), Imlay City (I) or West Bloomfield (W)
areas. During the year 2015, 15% of all Detroit families move to West Bloomfield area and 5%
move to Imlay City area. Also, 6% of all West Bloomfield families move to Detroit area and 4%
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move to a Imlay City area. Finally, 4% of all Imlay City families move to Detroit area and 6%
move to West Bloomfield area.
a. If a family now lives in Detroit area, what is the probability that the family will live in an
Detroit area two years from now.
Hint: First set up the Probability distribution matrix ?p? and then use it to solve the problem.

p=

D

W

I

D

x1,1

x2,1

x3,1

W

x21

x22

x23

I

x31

x32

x33

Where;
Xij = percentage of families moving from city to city in Michigan
b. Suppose that at present, 40% of all families live in Detroit area, 35% live in West
Bloomfield area, and 25% live in Imlay City Area. Two years from now, what percent of
Michigan families will live in Detroit area?
c. What problem might occur if this model was used to predict the future population
distribution of the state of Michigan?

Problem 4 The time between buses follows the mass function shown in Table 2 below. What is
the average length of time one must wait for a bus. Also draw the waiting time Paradox line
graph
Table 2: Mass Function
Time between
Busses

Probability

30 Minutes

1/4

1 Hour

1/4

2 Hours

?

a. There are four sections of 10th grade class at Walled Lake High School in Michigan. The
number in each section is as follows Section 1, 20 students; Section 2, 25 Students;
Section 3, 35 Students; and Section 4, 40 Students.
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i.
ii.

What is the average size of 10th grade section?
Suppose the high school randomly selects a Walled lake 10th grader that is a female.
On the average, how many students will be in her class

Problem 5
Each Airline passenger and his or her luggage must be checked to determine whether he or she
is carrying weapons onto the airplane. Suppose that at Detroit City Airport, an average of 10
passengers per minute arrive (inter-arrival time are exponential). To check passengers for
weapons the airport must have a checkpoint consisting of a metal detector and baggage X-ray
machine. Whenever a checkpoint is in operation, two employees are required. A checkpoint can
check an average of 12 passengers per minute (The time to check a passenger is exponential).
Under the assumption that the airport has only one checkpoint, answer the following questions.
a. What is the probability that a passenger will have to wait before being checked for
weapons?
b. On the average, how many passengers are waiting in line to enter the checkpoint?
c. On the average, how long will a passenger spend at a checkpoint?

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Solution details: