Managerial Problem Solving
CBTL212
Chapter 1
Introduction to Modeling
1.1 Introduction
• purpose of course is to
1. expose you to a variety of problems that have
been solved successfully with management
science methods and to
2. give you experience in modeling these
problems in the Excel spreadsheet package.
Introduction cont’
• Key to virtually every management science
application is a mathematical model.
• a mathematical model is a quantitative
representation, or idealization, of a real
problem.
• Represented or expressed in terms of
mathematical expressions (equations and
inequalities) or as a series of interrelated
cells in a spreadsheet.
Introduction cont’
• Purpose of a mathematical model represent the essence of a problem in a
concise form.
This has several advantages:
1. it enables a manager to understand the
problem better [scope, solutions and data
requirements]
Introduction cont’
2. allows analysts to employ a variety of the
mathematical solution procedures that
have been developed over the past halfcentury. [ usually computer intensive]
3. the modeling process itself, if done
correctly, often helps to “sell” the solution
to the people who must work with the
system that is eventually implemented.
1.2 A Waiting-Line Example
• A mathematical model is a set of
mathematical relationships that represent, or
approximate, a real situation.
• Descriptive models - Models that simply
describe a situation
• Optimization models - models that suggest
a desirable course of action
A Descriptive Model
Example: Waiting line or Queueing problem
• Consider a shop with a single cash register
(till).
• The manager first wants to build a model
that reflects the current situation at the
store. (descriptive model)
• Later, he will alter the model to predict
what might make the situation better.
(optimization model)
A Descriptive Model cont’
Single Line Waiting System
Customer gets served at the till
Customer
arrives
Waiting line
Customer leaves
A Descriptive Model cont’
• To describe the current situation, the
manager realizes that there are two
important inputs to the problem:
1. the arrival rate of potential customers to
the store and
2. the rate at which customers can be served
by the single cashier.
A Descriptive Model cont’
• Logic behind problem:
– as the arrival rate increases and/or (input)
– the service rate decreases, (input)
– the waiting line will tend to increase and
(output)
– each customer will tend to wait longer in line
(output)
A Descriptive Model cont’
• By making several simplifying assumptions
about the nature of the arrival and service
process at the store we can relate the inputs
to the outputs using a mathematical model
i.e. an equation
A Descriptive Model cont’
• A is the arrival rate of customers per
minute,
• S is the service rate of customers per
minute, and
• W is the average time a typical customer
waits in line
A Descriptive Model cont’
• We must determine how the manager can
obtain the inputs he needs. There are
actually three inputs:
– (1) the arrival rate A, (stopwatch)
– (2) the service rate S, and (stopwatch)
– (3) the number in the store, labeled N, that will
induce future customers not to enter.
(observation)
A Descriptive Model cont’
Descriptive queueing model for convenience
store
Inputs
Arrival rate (customers per minute)
Service rate (customers per minute)
Maximum customers (before others go elsewhere)
Outputs
Average number in line
Average time (minutes) spent in line
Percentage of potential arrivals who don't enter
0.5
0.4
5
2.22
6.09
27.1%
A Descriptive Model cont’
• These values indicate:
– that slightly more than 2 customers are waiting
in line on average,
– an average customer waits slightly more than 6
minutes in line, and
– about 27% of all potential customers do not
enter the store at all (due to the perception that
waiting times will be long).
A Descriptive Model cont’
• The power of the model is that it allows
the manager to ask many What-If
questions.
• For example, what if he could somehow
speed up the cashier, say, from 2.5 minutes
per customer to 1.8 minutes per customer?
– 1/1.8 = 0.556 customers per minute
A Descriptive Model cont’
Descriptive queueing model for convenience
store
Inputs
Arrival rate (customers per minute)
Service rate (customers per minute)
Maximum customers (before others go
elsewhere)
Outputs
Average number in line
Average time (minutes) spent in line
Percentage of potential arrivals who don't enter
0.5
0.556
5
1.41
3.22
12.6%
A Descriptive Model cont’
• what if he could somehow speed up the
cashier, say, from 2.5 minutes per customer
to 1.25 minutes per customer?
– 1/1.25 = 0.8 customers per minute
A Descriptive Model cont’
Descriptive queueing model for convenience store
Inputs
Arrival rate (customers per minute)
Service rate (customers per minute)
Maximum customers (before others go elsewhere)
0.5
0.8
5
Outputs
Average number in line
Average time (minutes) spent in line
0.69
1.42
Percentage of potential arrivals who don't enter
3.8%
A Descriptive Model cont’
• As the manager increases the service rate,
the output measures improve more than he
might have expected
• the manager should examine the
reasonableness of the assumptions – test it!!
An Optimization Model
• A model that suggest a desirable course of
action
• Reflects economic information, such as:
– the cost of speeding up service,
– the cost of making customers wait in line, or
– the cost of losing customers.
An Optimization Model cont’
• Assume that the manager can do one of
three things:
– Decision (1) leave the system as it is,
– Decision (2) hire a second person to help the
first cashier process customers more quickly
• average service time goes from 2.5 to 1.8 minutes
– Decision (3) lease a new model of cash register
that will speed up the service process
significantly.
• Average service time goes from 2.5 to 1.25 minutes
An Optimization Model cont’
• He needs to examine three types of costs:
1. The first is the cost of hiring the extra
per son or leasing the new cash register.
– hourly wage for the extra person is $8, and
the
– cost to lease a new cash register (converted
to a per-hour rate) is $11 per hour.
An Optimization Model cont’
2. The second type of cost is the “cost” of
making a person wait in line.
– a customer who has to wait a long time might
not return.
– This cost is difficult to estimate on a perminute or per-hour basis, but we assume it’s
approximately $13 per customer per hour
in line.
An Optimization Model cont’
3. The opportunity cost for customers who
decide not to enter the store.
– The store loses not only their current revenue
but also potential future revenue if they decide
never to return.
– Lets assume it’s approximately $25 per lost
customer.
1
2
3
4
5
Decision queueing model for convenience store
Inputs
Arrival rate (customers per minute)
Service rate (customers per minute)
Maximum customers (before others go
6 elsewhere)
7
8 Cost of extra person per hour
9 Cost of leasing new cash register per hour
10 Cost per customer per hour waiting in line
11 Cost per customer who doesn't enter the store
12
13 Outputs
14 Average number in line
15 Average time (minutes) spent in line
16 Percentage of potential arrivals who don't enter
17
18 Cost information
19 Cost of extra person per hour
20 Cost of leasing new cash register per hour
21 Cost per hour of waiting time
22 Cost per hour of lost customers
23
24 Total cost per hour
Decision 1
0.5
0.4
Decision 2
0.5
0.556
Decision 3
0.5
0.8
5
5
5
$0
$0
$13
$25
$8
$0
$13
$25
$0
$11
$13
$25
2.22
6.09
27.1%
1.41
3.22
12.6%
0.69
1.42
3.8%
$0
$0
$28.87
$203.29
$8
$0
$18.31
$94.52
$0
$11
$8.91
$28.52
$232.16
$120.82
$48.43
1.3 MODELING VERSUS
MODELS
• Learning specific models is essentially a
memorization process—memorizing the
details of a particular model, such as the
transportation model, and possibly learning
how to “trick” other problems into looking
like a transportation model.
• Modeling, on the other hand, is a process,
where you abstract the essence of a real
problem into a model, spreadsheet or
1.4 THE SEVEN-STEP
MODELING PROCESS
Step 1: Problem Definition
– The analyst first defines the organization’s
problem.
– Defining the problem includes specifying the
organization’s objectives and the parts of the
organization that must be studied before the
problem can be solved
1.4 THE SEVEN-STEP
MODELING PROCESS cont’
Step 2: Data Collection
– the analyst collects data to estimate the value
of parameters that affect the organization’s
problem
– gather exactly the right data and put the data
into an appropriate and consistent format for
use in the model.
•
•
•
asking questions of key people (such as the
accountants) throughout the organization,
studying existing organizational databases, and
performing time-consuming observational studies
of the organization’s processes
1.4 THE SEVEN-STEP
MODELING PROCESS cont’
Step 3: Model Development
• Analytical models - Models such as the
equation for W, where you use an equation
to relate inputs such as A and S to outputs
such as
• Simulation model (for very complex
problems), which enables you to
approximate the behavior of the actual
system W
1.4 THE SEVEN-STEP
MODELING PROCESS cont’
Step 3: Model Development cont’
– Most good models (where “good” really means
useful) capture the essence of the problem
without getting bogged down in less important
details.
– approximations of the real world, not mirror
images in every last detail.
1.4 THE SEVEN-STEP
MODELING PROCESS cont’
Step 4: Model Verification
– analyst now tries to determine whether the
model developed in the previous step is an
accurate representation of reality.
– A first step in determining how well the model
fits reality is to check whether the model is
valid for the current situation
Step 4: Model Verification cont’
• 2 Causes for unexpected outputs
1. the model could simply be a poor
approximation of the actual situation.
•
In this case, the analyst must refine the model until
it lines up more accurately with reality.
2. the model might be fine, but the analyst’s
intuition is not very good – provide wrong
prediction
1.4 THE SEVEN-STEP
MODELING PROCESS cont’
Step 5: Optimization and Decision Making
– Given a model and a set of possible decisions,
the analyst must now choose the decision or
strategy that best meets the organization’s
objectives
– maximizing profit or minimizing cost.
1.4 THE SEVEN-STEP
MODELING PROCESS cont’
Step 6: Model Communication to
Management
– The analyst presents the model and the
recommendations from the previous step to the
organization
– the analyst might present several alternatives
and let the organization choose the best one.
– The best strategy for successful presentation is
to involve key people in the organization
– make the model as intuitive and user-friendly as
possible
1.4 THE SEVEN-STEP
MODELING PROCESS cont’
Step 7: Model Implementation
– If the organization has accepted the validity and
usefulness of the study, the analyst then helps
to implement its recommendations.
– The implemented system must be monitored
constantly (and updated dynamically as the
environment changes) to ensure that the model
enables the organization to meet its objectives.
1.6 WHY STUDY
MANAGEMENT SCIENCE?
• Management science is an important area
and highly trained analysts are needed to
solve the large and complex problems faced
by the business world
1.6 WHY STUDY MANAGEMENT
SCIENCE? cont’
The following are some of the reasons for this
new-found relevance:
The modeling approach emphasized
throughout this book is an important way to
think about problems in general, not just the
specific problems we discuss.
This approach forces you to think logically
1.6 WHY STUDY MANAGEMENT
SCIENCE? cont’
Management science is admittedly built
around quantitative skills—it deals
primarily with numbers and relationships
between numbers
No matter what your spreadsheet abilities
are when you enter this course, by the time
you’re finished, you’ll be a proficient
spreadsheet user. .
1.6 WHY STUDY MANAGEMENT
SCIENCE? cont’
Management science modeling helps you
develop your intuition, and it also indicates
where intuition alone sometimes fails.