It takes 3 minutes to load and 2 minutes to unload a machine. Inspection and packing...
It takes 3 minutes to load and 2 minutes to unload a machine. Inspection and packing times total 1 minute; travel between machines takes 2 minutes. Machines run automatically for 20 minutes. Operators cost $10 per hour; machines cost $30 per hour. Document all answers completely. Suppose 12 machines are needed to meet the production requirement.
a) Which of the following assignments will minimize the total cost: {4, 4, 4} or {3, 4, 5}?
b) Is there another combination that will be less expensive than either of these two? An explanation/demonstration is required.
Homework Answers
Calculate the ideal number of identical machines assigned to an
operator 
.
…… (1)
Here, 
is concurrent activity time, 
is the independent machine activity
time, and 
is the independent operator activity
time.
Calculate the concurrent activity time 
.
Substitute 
for loading time of machine, and
for unloading time of machine.
Substitute 
for 
, 
for 
, and 
for 
in Equation (1), to calculate the
maximum number of machines assigned to an operator.
Since fractional number of machines cannot be assigned, take the
maximum number of machines 
that can be assigned to an operator as
4.
Therefore, the maximum number of machines 
that can be assigned to an operator
without creating machine idle time during a repeating cycle is
.
Calculate the ratio of cost per operator hour and cost per machine hour.
Here, cost per operator hour is 
, and cost per machine hour is
.
Substitute 
for 
, and 
for 
.
Write the condition for the machine assignment that minimizes the cost per unit produced.
Here, 
is the ratio of 
to 
.
Substitute 
for 
.
Substitute 
for
.
…… (2)
Substitute 0.4 for 
, 4 for 
, and 4.166 for 
in Equation (2).
Since the value of 
is less than 1, take maximum number of
machines 
that can be assigned to an operator
without creating machine idle time during a repeating cycle as
4.
Therefore, an assignment of 
will minimize the cost per unit produced.
Calculate the cost per unit produced 
when 4 machines are assigned to an operator.
...... (3)
Substitute 4 for 
, 
for 
, 
for 
, 4 for 
, 5 minutes for 
, 20 minutes for 
, and 
for 
in equation (3).
Therefore, the cost per unit produced when 4 machines are
assigned to an operator is 
.
For the optimum assignment equal to four, consider the case.
Case (1)
At 
: 
:
Substitute 
for 
.
…… (4)
Calculate the ideal number of identical machines assigned to an
operator 
.
Substitute 
for 
, and 
for 
.
Substitute 0.4 for 
, 4 for 
, and 
for 
in Equation (4).
Rewrite the equation to calculate the value of 
.
…… (5)
Case (2)
At 
: 
Substitute 0.4 for 
, 3 for 
, and 
for 
.
Rewrite the equation to calculate the value of 
.
…… (6)
Therefore, observe the Equations (5) and (6), and note the
optimum assignment of 4 machines occurs between the range of
.
Alternative 1
Let 15 machines needed to meet the production requirement be
divided in 3 parts as 
.
Calculate the repeating cycle time 
of 5 machines assigned to an
operator.
Here, 
is the number of identical machines
assigned an operator.
Substitute 5 for 
, 
for 
, and 
for 
.
Calculate the cost per unit produced 
.
Here, 
is the set of divided machine in
alternative 1, 
is the number of machine parts in
alternative 1, 
is the number of machines, and
is time 1 hour.
Substitute 
for 
, 3 for 
, 
for 
, 15 for 
, 
for 
, 30 minutes for 
, and 
for 
.
…… (7)
Alternative 2
Let 15 machines needed to meet the production requirement be
divided in 4 parts as 
.
As the majority of 4 number of machines assigned to an operator
is greater compared to 3 machines assigned to an operator, consider
the repeating cycle time 
of 4 machine to calculate the cost
per unit produced by alternative 2.
Calculate the repeating cycle time 
of 4 machines assigned to an
operator.
Here, 
is the number of identical machines
assigned an operator.
Substitute 4 for 
, 
for 
, and 
for 
.
Calculate the cost per unit produced 
.
Here, 
is the set of divided machine in
alternative 2, and 
is the number of machine parts in
alternative 2.
Substitute 
for 
, 4 for 
, 
for 
, 15 for 
, 
for 
, 24 minutes for 
, and 
for 
.
…… (8)
Alternative 3
Let 15 machines needed to meet the production requirement be
divided in 3 parts as 
.
Calculate the repeating cycle time 
of 6 machines assigned to an
operator.
Here, 
is the number of identical machines
assigned an operator.
Substitute 6 for 
, 5 minutes for 
, and 
for 
.
Calculate the average repeating cycle time 
for alternative 3.
Substitute 
for 
, 
for 
, and 
for 
.
Calculate the cost per unit produced 
.
…… (9)
Here, 
is the set of divided machine in
alternative 3, and 
is the number of machine parts in
alternative 3.
Substitute 
for 
, 3 for 
, 
for 
, 15 for 
, 
for 
, 30 minutes for 
, and 
for 
in equation (9).
…… (9)
Therefore, from Equations (7), (8), and (10), it is observed
that machine assignment of 
is the least costly assignment of 15 machines
that gives minimal cost for per unit of production.
Hence, assign the 15 machines in the alternative form of
to minimize cost per unit produced.
Calculate the ideal number of identical machines assigned to an
operator 
.
…… (1)
Here, 
is concurrent activity time, 
is the independent machine activity
time, and 
is the independent operator activity
time.
Calculate the concurrent activity time 
.
Substitute 
for loading time of machine, and
for unloading time of machine.
Substitute 
for 
, 
for 
, and 
for 
in Equation (1), to calculate the
maximum number of machines assigned to an operator.
Since fractional number of machines cannot be assigned, take the
maximum number of machines 
that can be assigned to an operator as
4.
Therefore, the maximum number of machines 
that can be assigned to an operator
without creating machine idle time during a repeating cycle is
.
Calculate the ratio of cost per operator hour and cost per machine hour.
Here, cost per operator hour is 
, and cost per machine hour is
.
Substitute 
for 
, and 
for 
.
Write the condition for the machine assignment that minimizes the cost per unit produced.
Here, 
is the ratio of 
to 
.
Substitute 
for 
.
Substitute 
for
.
…… (2)
Substitute 0.4 for 
, 4 for 
, and 4.166 for 
in Equation (2).
Since the value of 
is less than 1, take maximum number of
machines 
that can be assigned to an operator
without creating machine idle time during a repeating cycle as
4.
Therefore, an assignment of 
will minimize the cost per unit produced.
Calculate the cost per unit produced 
when 4 machines are assigned to an operator.
...... (3)
Substitute 4 for 
, 
for 
, 
for 
, 4 for 
, 5 minutes for 
, 20 minutes for 
, and 
for 
in equation (3).
Therefore, the cost per unit produced when 4 machines are
assigned to an operator is 
.
For the optimum assignment equal to four, consider the case.
Case (1)
At 
: 
:
Substitute 
for 
.
…… (4)
Calculate the ideal number of identical machines assigned to an
operator 
.
Substitute 
for 
, and 
for 
.
Substitute 0.4 for 
, 4 for 
, and 
for 
in Equation (4).
Rewrite the equation to calculate the value of 
.
…… (5)
Case (2)
At 
: 
Substitute 0.4 for 
, 3 for 
, and 
for 
.
Rewrite the equation to calculate the value of 
.
…… (6)
Therefore, observe the Equations (5) and (6), and note the
optimum assignment of 4 machines occurs between the range of
.
Alternative 1
Let 15 machines needed to meet the production requirement be
divided in 3 parts as 
.
Calculate the repeating cycle time 
of 5 machines assigned to an
operator.
Here, 
is the number of identical machines
assigned an operator.
Substitute 5 for 
, 
for 
, and 
for 
.
Calculate the cost per unit produced 
.
Here, 
is the set of divided machine in
alternative 1, 
is the number of machine parts in
alternative 1, 
is the number of machines, and
is time 1 hour.
Substitute 
for 
, 3 for 
, 
for 
, 15 for 
, 
for 
, 30 minutes for 
, and 
for 
.
…… (7)
Alternative 2
Let 15 machines needed to meet the production requirement be
divided in 4 parts as 
.
As the majority of 4 number of machines assigned to an operator
is greater compared to 3 machines assigned to an operator, consider
the repeating cycle time 
of 4 machine to calculate the cost
per unit produced by alternative 2.
Calculate the repeating cycle time 
of 4 machines assigned to an
operator.
Here, 
is the number of identical machines
assigned an operator.
Substitute 4 for 
, 
for 
, and 
for 
.
Calculate the cost per unit produced 
.
Here, 
is the set of divided machine in
alternative 2, and 
is the number of machine parts in
alternative 2.
Substitute 
for 
, 4 for 
, 
for 
, 15 for 
, 
for 
, 24 minutes for 
, and 
for 
.
…… (8)
Alternative 3
Let 15 machines needed to meet the production requirement be
divided in 3 parts as 
.
Calculate the repeating cycle time 
of 6 machines assigned to an
operator.
Here, 
is the number of identical machines
assigned an operator.
Substitute 6 for 
, 5 minutes for 
, and 
for 
.
Calculate the average repeating cycle time 
for alternative 3.
Substitute 
for 
, 
for 
, and 
for 
.
Calculate the cost per unit produced 
.
…… (9)
Here, 
is the set of divided machine in
alternative 3, and 
is the number of machine parts in
alternative 3.
Substitute 
for 
, 3 for 
, 
for 
, 15 for 
, 
for 
, 30 minutes for 
, and 
for 
in equation (9).
…… (9)
Therefore, from Equations (7), (8), and (10), it is observed
that machine assignment of 
is the least costly assignment of 15 machines
that gives minimal cost for per unit of production.
Hence, assign the 15 machines in the alternative form of
to minimize cost per unit produced.
Calculate the ideal number of identical machines assigned to an
operator 
.
…… (1)
Here, 
is concurrent activity time, 
is the independent machine activity
time, and 
is the independent operator activity
time.
Calculate the concurrent activity time 
.
Substitute 
for loading time of machine, and
for unloading time of machine.
Substitute 
for 
, 
for 
, and 
for 
in Equation (1), to calculate the
maximum number of machines assigned to an operator.
Since fractional number of machines cannot be assigned, take the
maximum number of machines 
that can be assigned to an operator as
4.
Therefore, the maximum number of machines 
that can be assigned to an operator
without creating machine idle time during a repeating cycle is
.
Calculate the ratio of cost per operator hour and cost per machine hour.
Here, cost per operator hour is 
, and cost per machine hour is
.
Substitute 
for 
, and 
for 
.
Write the condition for the machine assignment that minimizes the cost per unit produced.
Here, 
is the ratio of 
to 
.
Substitute 
for 
.
Substitute 
for
.
…… (2)
Substitute 0.4 for 
, 4 for 
, and 4.166 for 
in Equation (2).
Since the value of 
is less than 1, take maximum number of
machines 
that can be assigned to an operator
without creating machine idle time during a repeating cycle as
4.
Therefore, an assignment of 
will minimize the cost per unit produced.
Calculate the cost per unit produced 
when 4 machines are assigned to an operator.
...... (3)
Substitute 4 for 
, 
for 
, 
for 
, 4 for 
, 5 minutes for 
, 20 minutes for 
, and 
for 
in equation (3).
Therefore, the cost per unit produced when 4 machines are
assigned to an operator is 
.
For the optimum assignment equal to four, consider the case.
Case (1)
At 
: 
:
Substitute 
for 
.
…… (4)
Calculate the ideal number of identical machines assigned to an
operator 
.
Substitute 
for 
, and 
for 
.
Substitute 0.4 for 
, 4 for 
, and 
for 
in Equation (4).
Rewrite the equation to calculate the value of 
.
…… (5)
Case (2)
At 
: 
Substitute 0.4 for 
, 3 for 
, and 
for 
.
Rewrite the equation to calculate the value of 
.
…… (6)
Therefore, observe the Equations (5) and (6), and note the
optimum assignment of 4 machines occurs between the range of
.
Alternative 1
Let 15 machines needed to meet the production requirement be
divided in 3 parts as 
.
Calculate the repeating cycle time 
of 5 machines assigned to an
operator.
Here, 
is the number of identical machines
assigned an operator.
Substitute 5 for 
, 
for 
, and 
for 
.
Calculate the cost per unit produced 
.
Here, 
is the set of divided machine in
alternative 1, 
is the number of machine parts in
alternative 1, 
is the number of machines, and
is time 1 hour.
Substitute 
for 
, 3 for 
, 
for 
, 15 for 
, 
for 
, 30 minutes for 
, and 
for 
.
…… (7)
Alternative 2
Let 15 machines needed to meet the production requirement be
divided in 4 parts as 
.
As the majority of 4 number of machines assigned to an operator
is greater compared to 3 machines assigned to an operator, consider
the repeating cycle time 
of 4 machine to calculate the cost
per unit produced by alternative 2.
Calculate the repeating cycle time 
of 4 machines assigned to an
operator.
Here, 
is the number of identical machines
assigned an operator.
Substitute 4 for 
, 
for 
, and 
for 
.
Calculate the cost per unit produced 
.
Here, 
is the set of divided machine in
alternative 2, and 
is the number of machine parts in
alternative 2.
Substitute 
for 
, 4 for 
, 
for 
, 15 for 
, 
for 
, 24 minutes for 
, and 
for 
.
…… (8)
Alternative 3
Let 15 machines needed to meet the production requirement be
divided in 3 parts as 
.
Calculate the repeating cycle time 
of 6 machines assigned to an
operator.
Here, 
is the number of identical machines
assigned an operator.
Substitute 6 for 
, 5 minutes for 
, and 
for 
.
Calculate the average repeating cycle time 
for alternative 3.
Substitute 
for 
, 
for 
, and 
for 
.
Calculate the cost per unit produced 
.
…… (9)
Here, 
is the set of divided machine in
alternative 3, and 
is the number of machine parts in
alternative 3.
Substitute 
for 
, 3 for 
, 
for 
, 15 for 
, 
for 
, 30 minutes for 
, and 
for 
in equation (9).
…… (9)
Therefore, from Equations (7), (8), and (10), it is observed
that machine assignment of 
is the least costly assignment of 15 machines
that gives minimal cost for per unit of production.
Hence, assign the 15 machines in the alternative form of
to minimize cost per unit produced.
Add Answer to:
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