“continuous or semi-continuous”
“intermittent”

Job Shop
Low Volume
(unique)
Medium Volume
(high variety)
High Volume
(lower variety)
Very high volume
(standardized)
CABG Surgery
Unit variable costs generally too high
Batch Process
Exec. Shirt
Manzana Insurance
Worker-paced line
Machine-paced line
Toshiba
Toyota
Continuous process
Utilization of fixed capital generally too low
National
Cranberry
• Categorizes processes into one of five clusters
• Similar processes tend to have similar problems
• There exists a long-term drift from the upper left to the lower right

 Form a group of 2-3 students
 From your experience/observation, select a product produced for each of these processing models:
 Job shop
 Batch processing
 Assembly line
 Continuous processing
 Share results with the class

 Inventory (WIP in a process)
 Flow time
First choose an appropriate flow
Unit – a customer, a car, a scooter, etc.
 Time it takes a unit to get through the process
 Flow rate (throughput rate)
 Rate at which the process is delivering output
 Maximum rate that a process can generate supply is called the capacity of the process
Process
Inputs
Outputs
Flow units
(raw material, customers)

Topic Flow Unit
U.S. Immigration Applications
Flow Rate Flow Time Inventory
Approved/rejected cases
(6.3 MM/year)
Average processing time (7.6 months)
Pending cases (4.0
MM)
Champagne Industry Bottles of Champagne 260 MM bottles per year 3.46 years in cellar 900 Million bottles
MBA Program MBA Student 600 students/year 2 years 1200 students
Muhlenberg College
Outback Steak House

What it is: Inventory (I) = Flow Rate (R) * Flow Time (T)
Implications:
• Out of the three performance measures (I,R,T), two can be chosen by management, the other is GIVEN by nature
• Hold throughput (flow rate) constant: Reducing inventory = reducing flow time
Patients
11
10
9
6
5
8
7
2
1
4
3
0
7:00
Flow Time
Inventory
Cumulative
Inflow
Cumulative
Outflow
8:00 9:00
Inventory =Cumulative Inflow – Cumulative Outflow
10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00
Time

 Suppose that from 12 to 1 p.m. 200 students per hour enter the GQ and each student is in the system for an average of 45 minutes. What is the average number of students in the
GQ?
 Inventory = Flow Rate * Flow Time
 = 200 per hour * 45 minutes (= 0.75 hours)
 = 150 students
 Problem 2.2: Airline check-in data indicate from 9 to 10 a.m. 255 passengers checked in. Moreover, based on the number waiting in line, airport management found that on average, 35 people were waiting to check in. How long did the average passenger have to wait?
 If ten students on average are waiting in line for sandwiches and each is in line for five minutes, on average, how many students are arrive each hour for sandwiches?
 Flow Rate = Inventory / Flow Time =
10 Students / 5 minutes = 0.083 hour
 = 120 students per hour
 Flow Time = Inventory / Flow Rate =
35 passengers / 255 passengers per hour = 0.137 hours
 = 8.24 minutes

Waiting occurs in
Service facility
 Fast-food restaurants
 post office
 grocery store
 bank
Manufacturing
Equipment awaiting repair
Phone or computer network
Product orders
Why is there waiting?

 Average number of customers waiting
 In the queue (L q
)
 In the system (L)
 Average time customers wait
 In the queue (W q
)
 In the system (W)
 System utilization ( r )

Single Server
Multiple
Servers
Multiple Single
Servers

1. Single server, exponential service time (M/M/1)
2. Multiple servers, exponential service time (M/M/s)
A Taxonomy
M / M / s
Poisson Arrival Exponential Number of
Distribution Service Dist Servers where
M = exponential distribution (“Markovian”)
(Both Poisson and Exponential are Markovian – hence the “M” notation)

Given l m s
Calculate
L q
L
W q
W
P n r
=
=
= customer arrival rate service rate (1/m = average service time) number of servers
=
=
=
=
=
= average number of customers in the queue average number of customers in the system average waiting time in the queue average waiting time (including service) probability of having n customers in the system system utilization
Note regarding Little’s Law: L = l * W and Lq = l * Wq

Model 1: M/M/1 Example
The reference desk at a library receives request for assistance at an average rate of 10 per hour (Poisson distribution). There is only one librarian at the reference desk, and he can serve customers in an average of 5 minutes (exponential distribution). What are the measures of performance for this system? How much would the waiting time decrease if another server were added?
M/M/s Queueing Model Template l  m  s =
Data
10
12
1
(mean arrival rate)
(mean service rate)
(# servers)
Prob(W > t) = 0.135335
when t = 1
0
Prob(W q
> t) = 0.112779
when t = 1
Results
L =
L q
=
5 Number of customers in the system
4.166666667 Number of customers in the queue
0.5 Waiting time in the system
0.416666667 Waiting time in the queue
W =
W q
= r 
P
0
=
0.833333333 Utilization
0.166666667 Prob zero customers in the system

Beefy Burgers is considering changing the way that they serve customers. For most of the day (all but their lunch hour), they have three registers open. Customers arrive at an average rate of 50 per hour. Each cashier takes the customer’s order, collects the money, and then gets the burgers and pours the drinks. This takes an average of 3 minutes per customer (exponential distribution).
They are considering having just one cash register. While one person takes the order and collects the money, another will pour the drinks and another will get the burgers. The three together think they can serve a customer in an average of 1 minute. Should they switch to one register?

3 Slow Servers l  m  s =
Data
50
20
3
(mean arrival rate)
(mean service rate)
(# servers)
Prob(W > t) = 6.38E-05 when t = 1
0
Prob(W q
> t) = 4.34E-05 when t = 1
1 Fast Server
Results
L = 6.011235955 Number of customers in the system
L q
= 3.511235955 Number of customers in the queue
W = 0.120224719 Waiting time in the system
W q
= 0.070224719 Waiting time in the queue r  0.833333333 Utilization
P
0
= 0.04494382 Prob zero customers in the system l  m  s =
Data
50
60
1
(mean arrival rate)
(mean service rate)
(# servers)
Prob(W > t) = 4.54E-05 when t = 1
0
Prob(W q
> t) = 3.78E-05 when t = 1
Results
L = 5 Number of customers in the system
L q
= 4.166666667 Number of customers in the queue
W = 0.1 Waiting time in the system
W q
= 0.083333333 Waiting time in the queue r  0.833333333 Utilization
P
0
= 0.166666667 Prob zero customers in the system
W is less for one fast server, so choose this option.

We can use the results from queuing theory to make the following types of decisions:
Cost
How many servers to employ
Whether to use one fast server or a number of slower servers
Whether to have general purpose or faster specific servers
Total Cost
Cost of
Service Capacity
Cost of customers waiting
Optimum
Service Capacity
Goal: Minimize total cost = cost of servers + cost of waiting

 Cost of service: # Servers *
Cost of each server
 Service cost = s * C s
 Cost of Waiting: Cost of waiting
* Time waiting * number of customers/time unit
 Waiting Cost = l * C w
* W
 If you save more in waiting than you spend in service, make the change
 Example
 A fast food restaurant has three servers, each earning $10 per hour. Fifty customers per hour arrive and a server can serve a customer in three minutes.
Should the restaurant add a fourth server if the cost of a customer waiting is estimated at
$20 per hour?
 Answer
Service Cost
Current Proposed Difference
$ 30.00
Waiting Cost $ 120.22
$
$
40.00
60.66
$
$
10.00
(59.56)

The Southern Railroad Company has been subcontracting for painting of its railroad cars as needed. Management has decided the company might save money by doing the work itself. They are considering two alternatives. Alternative 1 is to provide two paint shops, where painting is to be done by hand (one car at a time in each shop) for a total hourly cost of $70. The painting time for a car would be 6 hours on average
(assume an exponential painting distribution) to paint one car.
Alternative 2 is to provide one spray shop at a cost of $175 per hour.
Cars would be painted one at a time and it would take three hours on average (assume an exponential painting distribution) to paint one car.
For each alternative, cars arrive randomly at a rate of one every 5 hours. The cost of idle time per car is $150 per hour.
 Estimate the average waiting time in the system saved by alternative 2.
 What is the expected total cost per hour for each alternative? Which is the least expensive?
Answer: Alt 2 saves 1.87 hours. Cost of Alt 1 is: $421.25 / hour and cost of Alt 2 is
$400.00 /hour.

Annual inventory costs (as a % of item value) include financing costs, depreciation, obsolescence, storage, handling, theft
 Obtaining data
 Look up inventory value on the balance sheet
 Look up cost of goods sold (COGS) from earnings statement – not sales!!
 Common benchmark is inventory turns
 Inventory Turns = COGS/ Inventory Value
 Compute per unit inventory costs:
Per unit inventory costs =
Annual inventory costs (as a % of item cost) /
Inventory turns

 Problem 2.3: A manufacturing company producing medical devices reported $60 million in sales last year. At the end of the year, they had $20 million worth of inventory in ready-to ship devices.
 Assuming that units are valued at $1000 per unit and sold at $2000 per unit, what is the turnover rate?
 Assume the company uses a 25% per year cost of inventory. What is the inventory cost for a $1000 (COGS) item. Assume that inventory turns are independent of price.
 Answer
Sales = $60,000,000 per year / $2000 per unit = 30,000 units sold per year @ $1000
COGS per unit
Inventory = $20,000,000 / $1000 per unit =
20,000 units in inventory
Turns = COGS/Inventory =
$30,000,000/$20,000,000 = 1.5 turns
Cost of Inventory: For a $1000 product, the total inventory cost (for one turn) is
$1000* 25% or $250. This divided by 1.5 turns gives an absolute inventory cost of
$166.66.

 Pipeline inventory
Cumulative patients
6
5
4
3
1.5 hours
1.5 Patients
2
7:00 8:00 9:00
1
 Seasonal Inventory
10:00 11:00 12:00
Time
(In 000 ’s)
Figure 2.10: Seasonal inventory - Sugar

 Cycle Inventory
 Decoupling inventory/Buffers
 Safety Inventory
Cumulative
1200
Inflow and outflow 1000
Cumulative inflow
800
600
Safety inventory
400
200
0
Cumulative outflow
Days of the month
Figure 2.12
: Safety inventory at a blood bank

Industries with higher gross margins tend to have lower inventory turns
Retail
Hardware stores: 3.5
Retail Nurseries & Garden Supply:
3.3
General Merchandise Stores: 4.7
Grocery Stores: 12.7
New & Used Car Dealers: 6.8
Gas stations & mini-marts: 39.3
Apparel & Accessories: 3.5
Furniture & home furnishings: 4.1
Drug Stores: 5.3
Liquor Stores: 6.6
Other Retail Stores: 4.3
Wholesale
Groceries & related: 17.8
Vehicles & automotive: 6.9
Furniture & fixtures: 5.5
Sporting goods: 4.8
Drug store items: 8.5
Apparel & related: 5.5
Petroleum & related: 42.4
Alcoholic beverages: 8.5
Source: Bizstats.com