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# What does utility rate measure? Description

##### Description: These notes cover the concepts that will be covered on the first midterm. Yellow highlight is for definitions of concepts, green highlight is for examples, and blue is equations. If you have any questions about something on here or would like to see examples I didn't include, feel free to email me at elaurien@usc.edu
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Midterm One Study Guide

## What does utility rate measure?

∙ Made to Order—product that isn’t made until after the order is placed

o This produces higher quality and prevents excess inventory, but is slower than made to  stock

o Use MTO for expensive and customizable items

▪ Luxury cars, airplanes

∙ Made to Stock—product that is made before an order is placed

o MTS is fast but may produce lower quality and inventory may be too high or you may  have to throw away some product

o Use MTS for cheap, uniform items

▪ McDonalds

Measurements for a Process

## How can one calculate capacity after cross-training?

∙ Capacity—the maximum number of units that can be processed per unit of time o Determined by the bottleneck—process with the smallest capacity

∙ Flow Rate (throughput rate)—the flow you actually process; average rate of inflows and  outflows expressed as a rate

∙ Utilization Rate—a measure of efficiency, shows what percentage of time your workers are  actually working

o ���������������������� �������� =�������� �������� (���������������� ��������)

����������������

o Starbucks Example

▪ Capacity of a cashier is 96 customers per shift

## What is the formula of little's law?

If you want to learn more check out What does feedforward regulation observe?

▪ Flow rate is 72 customers per shift

▪ Utilization rate = 72/96 = 75%

∙ This means that the cashier is busy 75% of the time, idle 25% of the time

before making managerial decisions

∙ Utilization rate cannot be more than 1. If demand is higher than the

capacity, some customers will have to go elsewhere or cannot be

served.

∙ Bottleneck Resource—process with the smallest capacity

o If you have one product (that goes through every station), the bottleneck is the slowest  workstation with the lowest capacity and highest utilization

o If you have multiple products, the bottleneck can be identified as the resource with the  highest utilization rate.

∙ Flow Time (throughput time)—the average time a unit stays in the system o Hamburger Example

▪ The first 10 burgers take 87 seconds to produce

∙ 60 seconds to cook + 27 seconds to assemble

▪ The second 10 burgers take 114 seconds to produceWe also discuss several other topics like What does the stubborn child law state?

∙ 60 seconds to cook + 27 seconds waiting (while first 10 are assembled) +

27 seconds to assemble

▪ Average out the two times to find the flow time

∙����+������

��= ������. �� �������������� = ��: ����. ��

∙ Work in Process (WIP)—the number of units in the system at a point in time being stored,  processed, or waiting.

Kristen’s Cookie Case—know the concepts, not just the facts We also discuss several other topics like What do the domains bacteria and archaea consist of?

∙ How many orders can you complete in four hours?

o Find the Bottleneck!

▪ Calculate the capacity of each resource

∙ The oven is being used for 10 minutes per order.

o Capacity = 6 orders/hour

o Capacity = 7.5 orders/hour

∙ Kristen’s roommate is working for setting the oven, packing and

paying—4 minutes/order

o Capacity = 15 orders/hour

▪ Because the oven has the lowest capacity, it is the bottleneck

o Every order after the first one will be waiting on the bottleneck (oven) which means  that it will take 10 additional minutes.

o Four hours * 60 minutes = 240 minutes If you want to learn more check out What are the stages of psychosocial development according to freud?

o 240 minutes – 26 minutes for the first order = 214 remaining minutes

o 214/10 minutes per order = 21.4 orders + 1 initial order = 22.4 orders

o Total = 22 orders in four hours

∙ Calculating orders in a certain amount of time

o �� +������������������ �������� ���������� �������������� ����������

�������� �������������������� ���������������� ������ �� ���������� = ���������������� ������ �� ��������������

∙ How much are we willing to pay for another oven?

o We work for four hours a day, 300 days a year If you want to learn more check out What do you call the people who own stock in a company?

o We currently produce at full capacity If you want to learn more check out How does implicit differentiation help people understand derivatives?

o We sell cookies for \$1.50 per dozen

o How much more could we produce with another oven?

▪ Find the new bottleneck!

∙ With two ovens, the ovens capacity is 12 orders/hour

∙ Kristen’s capacity is still 7.5 orders/hour so she is the new bottleneck

taking 8 minutes/order

▪ Use above calculation to determine how many orders we can produce with two  ovens

∙ �� +������−����

��= ����. ���� = ���� ������������ ���� �������� ����������

▪ We could produce 5 more orders a day with another oven

o How much more money could we make per year with another oven?

▪ This is the maximum that we would be willing to pay for another oven to make  it worth the expense

▪ Our profit is 80 cents/order

▪ We can sell 5 dozen more a day

▪ . ���� ∗ �� ∗ ������ = \$�������� �������������������� ������������

▪ We would be willing to pay \$1200 for another oven as long as there was  enough demand for this

∙ Flexibility—given demand, how much capacity do you need to meet the demand? ∙ Cross-training—each person is trained to do multiple jobs in order to increase capacity o To calculate Capacity after cross-training, take time each job takes and add them  together to represent the total amount of time that person can spend on each order (because they can do both jobs)

o Next, take the total minutes worked in an hour (60 * number of cross-trained workers)  and divide by the total time for both jobs to be completed

Calculating Capacity with Multiple Products (unable to meet demand)

∙ You must know demand to be able to determine the bottleneck

∙ To determine the capacity of a process:

▪ ���������������� ���� �������������� =���������������� ���� ��������������������

% ������������ ������ ��������������������

∙ % ������������ ������ �������������������� =�������� �������� ���� ��������������������

�������� �������� ���� ������������ ��������������

Little’s Law

�������� �������� =������

�������� ��������

∙ Flow time is given as an average

∙ Increasing Revenue

o Revenue is proportionate to flow rate, increasing flow rate increases revenue ▪ Increase FR by increasing WIP or decreasing flow time

∙ Practice Problem

o Call center employs 1000 agents—WIP

o Every month 125 leave and 125 are hired—flow rate

o How long on average does an agent work for this call center?

▪ �������� �������� =��������

������ = �� ������������ ���� ��������������

o Cost of hiring and training a new agent = \$2500

o Want to increase average work time to 20 months

o How much do the hiring and training costs decrease?

▪ What is the new flow rate if flow time is 20 months?

∙ ���� =��������

��

∙ �� = ���� ������������������ ������ ����������

▪ What is the current cost of training?

∙ 125 * 2500 = 312,500

▪ What would be the new cost of training?

∙ 50 * 2500 = 125,000

▪ Total Decrease = 312,500 – 125,000 = 187,500 savings per month

o What is the increase in salary the manager can afford to offer?

▪ Total savings is the additional amount the manager could afford per month ▪������������

��������∗��∗����= ��. ���� �������������������� ������ ��������

Waiting Line Management

∙ Uncertainty and Variability—as this increases, wait time increases

∙ Utilization—as utilization increases, wait time increases exponentially

∙ Risk Pooling—combining lines into one queue that leads to multiple servers o This causes the line to move faster because one problem doesn’t stop an entire line  from moving completely, and it ensures that someone who is in line first will get to a  server first

∙ Notations for wait line formula

o a = interarrival time

o 1/a = average arrival rate—flow rate

o p = average service time

o 1/p = average service rate

o m = number of servers

o CV = coefficient of variation

∙ Formulas for wait time

o ���������������������� = (��

��∗��)

o ������ ���� �������������� =����∗ �� = �� ∗ ��

o ������ ���� ���������� = ���� ∗����

o ������ =����(���� + ��)

o ���� =���������������� ������������������

��������

∙ Queue time with one server:

��−��∗��������+��������

o ���� =����

��

▪ The first half of the equation can also be expressed as ��∗��

1−��

∙ Gap Example

o 10 customers per hour for the checkout line

o Flow rate = 1/a = 10 customers/hour

o a = 6 minutes

o standard deviation of interarrival time = 5

o average service time for checkout: p = 5 min; 1/p = 12 customers/hour o standard deviation of service times = 1 min

o one cashier; m = 1

o What is the waiting time in line?

��−��∗ ((������+������)/��)

▪����

▪ = 9.18 average wait time

o What is the flow time?

▪ FT = Tq + p = 9.18 + 5 = 14.18

Multiple Server Systems

∙ Define a system by: arrival pattern/service pattern/# of servers

∙ Notations

o M—exponential

o G—general (any distribution)

o D—deterministic

∙ NOTE—CV in exponential distributions is always 1 because the mean is the same as the standard  deviation

∙ Queue time with multiple servers

��−��∗��������+��������

o ���� =����∗��(√��(��+��)−��

��

∙ Call Center Example

o 11 operators

o Arrival rate = 200 calls/hour

o 1/a = 60/200 = .3 minutes/call

o a = .3

o Operators can serve 20 calls/hour

o 1/p = 60/20 = 3 mins/call

o p = 3

o Inter-arrival and service times are exponentially distributed

▪ �� =��

.��∗����=. ������

▪ What is the average waiting time?

∙ ���� =������∗.������√��(����)−��

��−.������∗��+��

��=2.06 minutes wait time

▪ How many customers are waiting?

∙ �������� = ��. ���� ∗��.��= ��. ����

∙ Call center deterministic example

o Service time is constant, arrival is still exponential

o If service time is constant, there is no variability and CVp = 0

o ���� =������∗.������√��(����)−��

��−.������∗��+��

��= ��. ���� �������������� �������� ��������

o �������� = ��. ���� ∗��.��= ��. �� ��������������

Wait line Perceptions

∙ Perceived waiting time can matter more than actual waiting time when it comes to customer  satisfaction

∙ Satisfaction = Perception – Expectation

o To increase satisfaction, you can make people’s perception better (distract them in line)  or you can lower expectations (tell someone the wait is 30 minutes when it will really be  20)

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