Folie 1

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Subject Management decision making
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Purpose The purpose of the course is to enable
the students to understand the capabilities of
information technology and its role in decision
support in health and social care systems and to
provide them with understanding of quantitative
modeling by using of several decision science
approaches (linear programming, gaming, computer
simulation modeling etc.) to help management
sharpen judgment and effectiveness in
decision-making.
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• Learning objectives
• Develop a general understanding of the management
  science/operations research approach to decision
  making in health and social care organizations.
• Realize that quantitative applications begin with
  a problem situation.
• Obtain a brief introduction to quantitative
  techniques and their frequency of use in
  practice.
• Understand that managerial problem situations
  have both quantitative and qualitative
  considerations that are important in the decision
  making process.
• Learn about models in terms of what they are and
  why they are useful.
• Identify the step-by-step procedure that is used
  in most quantitative approaches to decision
  making.
• Obtain an introduction to microcomputer software
  packages and their role in quantitative
  approaches to decision making

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Structure of the course Chapter 1
Introduction Chapter 2 Linear programming
Chapter 10 Project scheduling
PERT/CPM Chapter 12 Waiting line
models Chapter 13 Simulation Chapter
14 Decision analysis Chapter 15 Multicriteria
decision problems Chapter 16 Forecasting
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Chapter 1 Introduction
Management is a process by which organizational
goals (outputs) are achieved through the use of
corporate resources (inputs). These
organizational decisions (processes) are
typically made by managers.

• A manager's role can be categorized into
• Interpersonal - figurehead, leader, liaison
• Informational - monitor, disseminator,
  spokesperson
• Decisional - entrepreneur, problem solver,
  resource coordinator, and negotiator

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Chapter 1 Introduction
A decision refers to a choice made between
alternatives. Decision making in organizations
can be classified into two broad categories
problem solving and opportunity exploitation.

• Why Managers Need the Support of Information
  Technology.  It is very difficult to make good
  decisions without valid, timely and relevant
  information.
• Number of alternatives to be considered is
  increasing.
• Many decisions are made under time pressure.
• Due to uncertainty in the decision environment,
  it is frequently necessary to conduct a
  sophisticated analysis.
• It is often necessary to rapidly access remote
  information.

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Chapter 1 Introduction
Discovery, communication and collaboration tools
provide indirect support to decision making,
however there are several other information
technologies used to directly support decision
making.

• Decision Support Systems (DSS) provide support
  primarily to analytical, quantitative types of
  decisions.
• Executive (Enterprise) Support Systems (ESS)
  support the informational roles of executives.
• Group Decision Support Systems supports managers
  and staff working in groups.
• Intelligent Systems

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Chapter 1 Introduction
Decision Process
Decision makers go through a fairly systematic
process.
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Chapter 1 Introduction
A model (in decision making) is a simplified
representation of reality. It is simplified
because reality is too complex to be copied and
there is no need to contain irrelevant details.

• The benefits of modeling in decision making are
• The cost of virtual experimentation is much lower
  than the cost of experimentation with a real
  system.
• Models allow for the simulated compression of
  time.
• Manipulating the model is much easier than
  manipulating the real system.
• The costs of mistakes are much lower in virtual
  experimentation.
• Modeling allows a manager to better deal with the
  uncertainty by introducing what-ifs and
  calculating the risks involved in specific
  actions.
• Mathematical models allow the analysis and
  comparison of a very large number of possible
  alternative solutions.
• Models enhance and reinforce learning and support
  training.

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Chapter 1 Introduction
Representation by models can be done at various
degrees of abstraction. Models are thus
classified into four groups according to their
degree of abstraction

• An Iconic or Scale model is a physical replica of
  a system.
• An Analog model does not look like the real
  system but behaves like it.
• A Mathematical (Quantitative) model describes the
  system with the aid of mathematics and is
  composed of three types of variables (decision,
  uncontrollable and result)
• A Mental models provides a subjective description
  of how a person thinks about a situation. The
  model includes beliefs, assumptions,
  relationships and flows of work as perceived by
  that individual.

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Chapter 1 Introduction
Models

• Mathematical model
• simplified quantitative
• representation of a real world
• Represents the essence of a problem

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Chapter 1 Introduction
Real world
Models
1 cat
Qualitative (iconic) model
Quantitative (mathematical) model
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Chapter 1 Introduction
The role of IT?

Why managers need the
support of IT?

number of alternatives

time pressure

sophisticated analysis

data

Can the managers job be
fully automated?
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Chapter 1 Introduction
Decision making ranges from simple to very
complex decisions that fall along a continuum
that ranges from structured to unstructured.
Structured processes refer to routine
repetitive problems with standard solutions.
While Unstructured are "fuzzy," complex problems
with no clear-cut solutions.
Unstructured
New Service
Semistructured
Reorder
Structured
Fulfillment
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Chapter 1 Introduction
Decision support system (DSS) is a computer-based
information system that combines models and data
in an attempt to solve semistructured and
unstructured problems with user involvement.
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Chapter 1 Introduction
Every DSS consists of at least data management,
user interface, model management components, and
the end users. A few also contain a knowledge
management component.

• A DSS data management subsystem contains all the
  data that flow from several sources, and are
  extracted prior to their entry into a DSS
  database or a data warehouse.
• A model management subsystem contains completed
  models (financial, statistical, management
  science, or other quantitative models), and the
  routines to develop DSSs applications.
• The user interface covers all aspects of the
  communications between a user and the DSS.
• The Users.  The person (manager, or the decision
  maker) faced with the problem or decision that
  the DSS is designed to support
• A knowledge-based or intelligent subsystem
  provides the expertise for solving some aspects
  of the problem, or the knowledge that can enhance
  the operation of the other DSS components.

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Chapter 1 Introduction
DSS Process
When users have a problem they evaluate it using
these processes.
Model
Data
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Seven-Step Modeling Process

• Step 1 Problem Definition - Define the problem
  including the objectives and the parts of the
  organization that must be studied.
• Step 2 Data Collection Collect the data to
  estimate the value of parameters that affect the
  organizations problem.
• Step 3 Model Development Develop an analytical
  or simulation model.
• Step 4 Model Verification Determine whether
  the model is an accurate representation of
  reality.
• Step 5 Optimization and Decision Making Given
  the model and a set of possible decisions, the
  analyst must choose the decision that best meets
  the organizations objectives.
• Step 6 Model Communication to Management The
  analyst presents the model and the
  recommendations to the organization.
• Step 7 Model Implementation If the
  organization accepts the model then the analysts
  assists with implementation. Implementation must
  be monitored constantly to ensure that the model
  enables the organization to meets its objectives.

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Methods used most frequently

• Generally
• Linear programming
• Integer programming
• Network models
• Simulation
• In health and social care systems
• Linear programming
• Project scheduiling
• Waiting line models
• Simulation
• Decision analysis
• Multicriteria decisions
• Forecasting

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Chapter 2 Linear programming

• Learning objectives
• Obtaining an overview of the kinds of problems
  linear programming has been used to solve.
• Learning how to develop linear programming models
  for simple problems.
• Being able to identify the special features of a
  model that make it a linear programming model.
• Learning how to solve two variable linear
  programming models by the graphical solution
  procedure.
• Understanding the importance of extreme points in
  obtaining the optimal solution.
• Being able to interpret the computer solution of
  a linear programming problem.

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Applications
Chapter 2 Linear programming

• resource allocation
• inventory management
• transport optimization
• portfolio selection
• work force assignments
• financial mix strategy
• blending problems
• data envelopment analysis
• revenue management

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Kinds of optimization
Chapter 2 Linear programming

• Linear optimization
• Linear optimization with integer variables
• Nonlinear optimization
• Multi-objective decision making

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Linear optimization
Chapter 2 Linear programming

• Mathematically LP involves
• optimizing a linear function
• subject to several linear constraints (expressed
  as linear inequalities or equalities)
• Modelling
• Decision variables
• Constraints
• Objective function
• During World War II (George Dantzig)

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Software
Chapter 2 Linear programming

• Many software packages for solving LP problems
  exist (Solver in MS Excel, LINDO...)
• Sensitivity analysis which provides additional
  information that is important for managers
• Our emphasis
• mathematical formulation of the problem
• interpretation of the results

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Chapter 2 Linear programming

• A simple maximization
• problem
• Golf bag manufacturing
• Objective
• Maximize the total profit
• Decision variables
• S number of standard bags
• D number of deluxe bags

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Chapter 2 Linear programming
Objective function Total profit contribution
10S 9D Maximization problem Max 10S
9D Constraints Total hours of cutting time used
lt hours of cutting time available 7/10S 1D
lt 630 Total hours of sewing time used lt hours
of sewing time available 1/2S 5/6D lt
600 Total hours of finishing time used lt hours
of finishing time available 1S 2/3D lt
708 Total hours of packing time used lt hours of
packing time available 1/10S 1/4D lt 135 Two
additional constraints S gt 0, D gt 0
(nonnegativity constraints)
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Chapter 2 Linear programming
Mathematical statement Max 10S 9D Subject to
(s.t.) 7/10S 1D lt 630 cutting 1/2S 5/6D
lt 600 sewing 1S 2/3D lt 708
finishing 1/10S 1/4D lt 135 packing S, D gt
0 This mathematical model is a linear
programming model or linear program (Objective
function, Constraints). LP has notting to do
with computer programming.
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Chapter 2 Linear programming
Linear equation Y aX b
b the value of the Y when X0 (intercept on Y
axis) (constant) a the slope of the line (a1
gtslope is 45degrees) (constant)
Y
Possible notations Y - b aX Y - aX b Y aX
b
b
X
0
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Chapter 2 Linear programming
7/10S 1D lt 630 cutting gt linear equation
7/10S 1D 630 constraint equation
D
Constraint line
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Chapter 2 Linear programming
D
Redundand constraint line
Feasible region
S
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Chapter 2 Linear programming
D
Max 10S 9D
Feasible region
S
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Chapter 2 Linear programming
System of two equations with two
variables 7/10S 1D 630 1S 2/3D
708 Solution D252, S540
D
10S 9D 7668
Optimal (graphical) solution (D252, S540) at an
extreme point of the feasible region
10S 9D 5400
252
10S 9D 3600
S
540
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Chapter 2 Linear programming
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Max 3X1 4X2 s.t. 2X1 3X2 lt 24 3X1 X2
lt 21 X1 X2 lt 9 X1, X2 gt0
Chapter 2 Linear programming
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Max 3X1 4X2 s.t. 2X1 3X2 lt 24 3X1 X2
lt 21 X1 X2 lt 9 X1, X2 gt0
Chapter 2 Linear programming
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Chapter 2 Linear programming
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Sensitivity
Chapter 2 Linear programming

• When the optimal solution is found
• How changes in a LP's parameters (objective
  function coefficients and right-hand sides)
  affect the optimal solution?
• Dual price for the constraint the amount by
  which the optimal value is improved if the right
  hand side of the constraint is increased by 1.
• Reduced costs indicates how much the objective
  function coefficient of each decision variable
  would have to increase (maximization problem)
  before it would be possible for that variable to
  assume a positive value in the optimal solution.
  If a decision variable is already positive in the
  optimal solution, its reduced cost is zero.

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Integer programming problem
Chapter 2 Linear programming

• Some or all of the variables are required to be
  nonnegative integers
• Many real-life situations may be formulated as
  IPs
• Integer variables
• Binary variables

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Chapter 10 Project Scheduling (PERT/CPM)
Project management Project Scheduling (PERT/CPM)
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Objectives
Chapter 10 Project Scheduling (PERT/CPM)

• Describe project management tools and how they
  are used
• Describe the steps used in project planning,
  scheduling, monitoring and controlling, and
  reporting
• Explain techniques for estimating task completion
  times and costs
• Describe various scheduling tools, including
  Gantt charts and PERT/CPM
• charts
• Calculate completion times, start dates, and end
  dates for a project
• Understand the reasons why projects sometimes fail

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Project Management Overview
Chapter 10 Project Scheduling (PERT/CPM)

• Project management is the process of planning,
  scheduling, monitoring and controlling, and
  reporting upon the development of an system
• The goal of project management is to deliver an
  system that is acceptable to users and is
  developed on time and within budget
• Project manager or project leader or
• Project coordinator

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Project Management Overview
Chapter 10 Project Scheduling (PERT/CPM)

• Project managers typically perform four main
  tasks
• Project planning
• Project scheduling
• Project monitoring and controlling
• Project reporting

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Chapter 10 Project Scheduling (PERT/CPM)
Project Management Overview

• Task or activity
• Event or milestone
• The project manager leads and coordinates the
  team, monitors events, and reports progress

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Project Management Overview
Chapter 10 Project Scheduling (PERT/CPM)

• Identifying Tasks
• One of the most important variables is the size
  of the project, because the amount of work does
  not relate directly to the size of the project
• If one project is twice the size of another
  project, the larger project will take more than
  twice as many resources to develop
• Six times as many relationships can mean more
  delay, misunderstanding, and difficulty in
  coordinating tasks
• The capabilities of project team members also
  affect time requirements
• A less experienced analyst usually will need more
  time to complete a task than an experienced team
  member will

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Project Planning
Chapter 10 Project Scheduling (PERT/CPM)

• Estimating Task Completion Time and Cost
• Person-days
• Some tasks can be divided evenly so it is
  possible to use different combinations of time
  and people, up to a point
• In most tasks, however, time and people are not
  interchangeable

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Project Planning
Chapter 10 Project Scheduling (PERT/CPM)

• Estimating Task Completion Time and Cost
• Best-case estimate (B)
• Probable-case estimate (P)
• Worst-case estimate (W)
• Weight
• Expected task duration
• (B4PW)
• 6

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Project Planning
Chapter 10 Project Scheduling (PERT/CPM)

• Factors Affecting Time and Cost Estimates
• Project size and scope
• IT resources
• Prior experience with similar projects or systems
• Constraints

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Overview of Project Scheduling
Chapter 10 Project Scheduling (PERT/CPM)

• Project scheduling involves the creation of a
  specific timetable
• Dependent task
• Must balance task time estimates, sequences, and
  personnel assignments
• Several graphical planning aids can help

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Project Scheduling with Gantt Charts
Chapter 10 Project Scheduling (PERT/CPM)

• Gantt Chart
• A detailed Gantt chart for a very large project
  might be quite complex and hard to understand
• Task groups
• Are not an ideal tool for controlling a complex
  project

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Project Scheduling with PERT/CPM Charts
Chapter 10 Project Scheduling (PERT/CPM)

• The Program Evaluation Review Technique (PERT)
• Critical Path Method (CPM)
• The important distinctions between the two
  methods have disappeared over time, and today the
  technique is called either PERT, or CPM, or
  PERT/CPM

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Project Scheduling with PERT/CPM Charts
Chapter 10 Project Scheduling (PERT/CPM)

• Overview of PERT/CPM
• PERT/CPM is called a bottom-up technique
• Project tasks
• Once you know the tasks, their duration, and the
  order in which they must be performed, you can
  calculate the time that it will take to complete
  the project

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Project Scheduling with PERT/CPM Charts
Chapter 10 Project Scheduling (PERT/CPM)

• PERT/CPM Chart Format
• Task box
• T (task duration, or time)
• ES (earliest start)
• EF (earliest finish) expected project duration
• LF (latest finish)
• LS (latest start)

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Project Scheduling with PERT/CPM Charts
Chapter 10 Project Scheduling (PERT/CPM)

• Task Patterns
• Sequential tasks
• Multiple successor tasks
• Concurrent task
• Predecessor task
• Successor task
• Multiple Predecessor Tasks

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Project Scheduling with PERT/CPM Charts
Chapter 10 Project Scheduling (PERT/CPM)

• Complex Task Patterns
• When various task patterns combine, you must
  study the facts carefully in order to understand
  the logical sequence of tasks
• A project manager must understand that project
  calculations will not be accurate unless the
  underlying task pattern is logically correct

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Project Scheduling with PERT/CPM Charts
Chapter 10 Project Scheduling (PERT/CPM)

• A PERT/CPM Example with Five Tasks
• Figure shows a PERT/CPM chart with five tasks
• You must calculate LS and LF
• Remember that LS and LF work just the opposite
  from ES and EF
• Next figure shows the final version with LS and
  LF entered

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Project Scheduling with PERT/CPM Charts
Chapter 10 Project Scheduling (PERT/CPM)

• Critical Path
• Slack time
• If any task along the critical path falls behind
  schedule, the entire project is delayed
• A critical path includes all tasks that are vital
  to the project schedule

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Project Scheduling with PERT/CPM Charts
Chapter 10 Project Scheduling (PERT/CPM)

• Transforming a Task List into a PERT/CPM Chart
• You must develop three versions
• Version 1 Basic Structure
• Version 2 Enter ES and EF Values
• Version 3 Add LF and LS Values
• After you enter the LS and LS figures, you will
  be able to identify the critical path

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Project Scheduling with PERT/CPM Charts
Chapter 10 Project Scheduling (PERT/CPM)

• Comparing Gantt Charts and PERT/CPM
• One significant advantage of PERT/CPM charts is
  that all individual tasks and dependencies are
  shown
• A PERT/CPM chart displays the critical path for
  the overall project and the slack time
• A Gantt chart offers a rapid overview
• PERT/CPM and Gantt charts are not mutually
  exclusive techniques

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Project Monitoring and Controlling
Chapter 10 Project Scheduling (PERT/CPM)

• Monitoring and Control Techniques
• The project manager must keep track of tasks and
  progress of team members, compare actual progress
  to the project plan, verify the completion of
  project milestones, and set standards and ensure
  that they are followed

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Project Monitoring and Controlling
Chapter 10 Project Scheduling (PERT/CPM)

• Maintaining a Schedule
• Maintaining a project schedule can be a
  challenging task
• The better the original plan, the easier it will
  be to control the project
• If enough milestones and frequent checkpoints
  exist, problems will be detected rapidly
• It is mathematically possible for a project to
  have more than one critical path

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Project Reporting
Chapter 10 Project Scheduling (PERT/CPM)

• Project Status Meetings
• Most project managers schedule regular status
  meetings with the entire project team
• Each team member updates the group and identifies
  any problems or delays
• The meetings also give the project manager an
  opportunity to update the entire group, seek
  input, and conduct brainstorming sessions

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Project Reporting
Chapter 10 Project Scheduling (PERT/CPM)

• Project Status Reports
• A project manager must report regularly to his or
  her immediate supervisor, upper management, and
  users
• Should explain what you are doing to handle and
  monitor the problem
• Most managers recognize that problems do occur on
  most projects it is better to alert management
  sooner rather than later

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Project Management Software
Chapter 10 Project Scheduling (PERT/CPM)

• Project Management Software
• Project Management Example Using Microsoft
  Project
• Create a Gantt chart showing the necessary
  information
• After you complete the Gantt chart, you decide to
  view the data in the form of a PERT chart

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Project Management Software

• Project Management Example Using Microsoft
  Project
• Network diagram
• Each task box contains the task description, task
  identification number, task duration, start date,
  and end date
• Project planning is a dynamic task and involves
  constant change

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Keys to Project Success
Chapter 10 Project Scheduling (PERT/CPM)

• Business Issues
• The major objective of every system is to provide
  a solution to a business problem or opportunity
• A system that falls short of business needs also
  produces problems for users and reduces morale
  and productivity

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Keys to Project Success
Chapter 10 Project Scheduling (PERT/CPM)

• Budget Issues
• Cost overruns typically result from one or more
  of the following
• Unrealistic estimates
• Failure to develop an accurate cost forecast
• Poor monitoring of progress and inadequate
  reaction to early signs of problems
• Schedule delays due to unanticipated factors
• Human resource factors

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Keys to Project Success
Chapter 10 Project Scheduling (PERT/CPM)

• Schedule Issues
• Problems with timetables and project milestones
  can indicate a failure to recognize task
  dependencies, confusing effort with progress,
  poor monitoring and control methods, personality
  conflicts among team members, or turnover of
  project personnel

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Keys to Project Success
Chapter 10 Project Scheduling (PERT/CPM)

• Successful Project Management
• When problems occur, the project managers
  ability to handle the situation becomes the
  critical factor
• Sometimes, when a project experiences delays or
  cost overruns, the system still can be delivered
  on time and within budget if several less
  critical requirements are trimmed

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Keys to Project Success
Chapter 10 Project Scheduling (PERT/CPM)

• Successful Project Management
• If a project is in trouble because of a lack of
  resources or organizational support, management
  might be willing to give the project more
  commitment and higher priority
• A typical response is to push back the completion
  date
• Option only if the original target date is
  flexible and the extension will not create
  excessive costs or other problems

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Toolkit Summary
Chapter 10 Project Scheduling (PERT/CPM)

• Project management is the process of planning,
  scheduling, monitoring and controlling, and
  reporting upon the development of an system
• Begins with identifying and planning all specific
  tasks or activities
• Can use graphical tools such as Gantt charts and
  PERT/CPM charts to assist in the scheduling
  process

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Toolkit Summary
Chapter 10 Project Scheduling (PERT/CPM)

• A project manager uses a variety of techniques to
  monitor, control, and report project tasks
• Every successful system must support business
  requirements, stay within budget, and be
  available on time

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Chapter 12 Waiting line models

• Learning objectives
• Be able to identify where waiting line problems
  occur and realize why it is important to study
  these problems.
• Know the difference between single-channel and
  multiple-channel waiting lines.
• Understand how the Poisson distribution is used
  to describe arrivals and how the exponential
  distribution is used to describe services times.

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Chapter 12 Waiting line models
Learning objectives 4. Learn how to use formulas
to identify operating characteristics of the
following waiting line models Single-channel
model with Poisson arrivals and exponential
service times Multiple-channel model with Poisson
arrivals and exponential service
times Single-channel model with Poisson arrivals
and arbitrary service times Multiple-channel
model with Poisson arrivals, arbitrary service
times, and no waiting 5. Know how to incorporate
economic considerations to arrive at decisions
concerning the operation of a waiting line.
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Chapter 12 Waiting line models
Server- order filling
Customer leaves
Customer arrivals
Waiting line
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Chapter 12 Waiting line models
Waiting lines in health care organizations can be
a serious problem and the money spent to reduce
these lines is often money well spent. Staffing
a centralized appointment scheduling department
in Lourdes hospital Lourdes Hospital decided to
use a centralized system to schedule appointments
for outpatients and inpatients and ambulatory
services requested by physicians, their staff,
hospital personnel, and patients. Several
queuing models, each for different time of the
day, have been solved. A new system was a big
success. The number of formal complaints fell
from three or four each week to less than one per
week.
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Chapter 12 Waiting line models
Introduction

• A fact of life is that everyone spends a great
  deal of time in queues.
• Why look at waiting lines?
• Want to examine existing system to quantify its
  operating characteristics.
• Want to learn how to make a system better or find
  the best system

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Chapter 12 Waiting line models

• Basic approach
• First objective
• Analysis of an existing system to quantify its
  operating characteristics by using two
• basic modeling approaches
• 1. analytical search for mathematical
  formulas that describe operating characteristics
• problem mathematical models are typically to
  complex to solve simplification with often
  unrealistic assumptions
• simulation allows us to analyze much more
  complex systems
• problem commercial software packages need to
  be used instead of spreadsheets

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Chapter 12 Waiting line models

• Basic approach
• Second objective
• Optimization, where we attempt to find the best
  system
• 1. analysis of each of several competing systems
  (analytically or by simulation)
• problem its difficult to estimate the cost of
  making a patient wait an extra 2 minutes in line
  - but necessary
• choice between the cost of waiting and the cost
  of more efficient service

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Chapter 3 Waiting line models
Elements of waiting line Models

• Characteristics of arrivals
• Interarrival time are the times between
  successive customer (patient) arrivals.
• Assume customers arrive one at a time and all
  have the same priority
• Assume there is no balking or reneging.
• Balking is when a customer decides not to wait in
  line.
• Reneging is when a customer gets in line and then
  changes their mind and leaves the line.
• Limited waiting room

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Elements of waiting line Models
Chapter 12 Waiting line models

• Service discipline is the rule that states which
  customer, from all who are waiting, goes into
  service next.
• First-come-first-served (FCFS) always assume
  this discipline
• Service-in-random-order (SRO)
• Last-come-last-served (LCLS)
• Shortest-processing-time (SPT)
• Service Characteristics
• Where customers join a single line and are then
  served by the first available tell, the servers
  or tellers are in parallel.
• A waiting line network is when each machine has
  its own service time distribution, and a typical
  part might have to wait in line behind any or all
  of the machines on its routing.

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Interarrival times are the times between
successive patient arrivals
Chapter 12 Waiting line models
Distribution of patient arrivals Poisson
probability distribution
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Chapter 12 Waiting line models
Mean The mean is the sum of a set of observations
divided by the number of the observations.
Standard deviation The standard deviation
represents the average deviation of each value
from the mean. Variance The variance is the
square of the standard deviation. Probability Pro
bability is a way of expressing uncertainty.
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Chapter 12 Waiting line models

• Discrete versus continuous
• A distribution is discrete if it has a finite
  number of positive values.
• A distribution is continuous if its possible
  values are essentially some continuum.
• Graphical Characteristics
• Discrete distribution is a series of spikes
• Continuous distribution is characterized by a
  density function, a smooth curve. Probabilities
  are found as areas under the density function.

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Chapter 12 Waiting line models

• Symmetric versus Skewed
• A distribution is symmetric if the distribution
  to the left of the point is a mirror image of the
  distribution to the right of the point.
  Otherwise, it is skewed.
• Skewed to the right (or positively skewed) if the
  longer tail is the right tail. Otherwise it is
  skewed to the left (or negatively skewed)

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Chapter 12 Waiting line models

• Bounded versus Unbounded
• A distribution is bounded if there are values A
  and B such that no possible value can be less
  than A or greater than B.
• A is then the minimum possible value, and the
  value of B is the maximum possible value.
• A distribution is unbounded if there are no such
  bounds.
• Possible for a distribution to be bounded in one
  direction but not the other.
• Positive (or Nonnegative) versus Unrestricted
• Special case of bounded distributions occurs with
  variables that are inherently positive (or
  possibly nonnegative).

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Chapter 12 Waiting line models

• Common Probability distributions
• A family of distributions has a common name,
  such as normal. Each member of the family is
  specified by one or more numerical parameters.
• Uniform distribution
• The uniform distribution is the flat equally
  likely distribution.

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Chapter 12 Waiting line models

• The RAND function is Excels building block
  function for generating random numbers.
• Enter the formula RAND( ) into any cell. Press
  F9 to make it randomly change.
• Numbers created by this function have two
  properties
• Uniform property all numbers between 0 and 1
  have the same chance of occurring
• Independence property Different numbers
  generated are probabilistically independent.
• Sometimes called pseudo-random numbers.
• Each successive random number follows the
  previous random number by a complex arithmetic
  operation.

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Chapter 12 Waiting line models

• A histogram, also called a frequency chart,
  indicates the number of observations in each of
  several user-defined categories.
• Create histograms with an Excel add-in such as
  _at_RISK
• Situations may require that random numbers stay
  fixed. A technique called freezing them can be
  used.
• RISKView is a separate software package that can
  be used to analyze probability distributions.

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Chapter 12 Waiting line models

• The interactive capabilities of RISKView, with
  its sliders, make it perfect for finding
  probabilities or percentiles for any given
  distribution.

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Chapter 12 Waiting line models

• A discrete distribution is useful for many
  situations.
• The RISKDISCRETE functions can be used to
  generate random numbers.
• Enter the formula RISKDISCRETE(valRange,progRange
  ) into any cell.
• valRange is the range where the possible values
  are stored
• probRange is the range where their probabilities
  are stored.

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Chapter 12 Waiting line models

• The normal distribution (familiar bell-shaped
  curve) is useful in simulation modeling as a
  continuous input distribution.
• It is not always the most appropriate
  distribution
• Symmetric drawback because skewed distribution
  is more realistic
• Allow negatives often not appropriate in many
  situations
• Two parameters are its mean and standard
  deviation.

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Chapter 12 Waiting line models

• Normally distributed random numbers will almost
  certainly be within three standard deviations of
  the mean.
• Random numbers can be generated with the _at_RISKs
  RISKNORMAL function.
• Enter the formula RISKNORMAL(Mean,Stdev)

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Chapter 12 Waiting line models

• A triangular distribution is similar to a normal
  distribution but it is more flexible and
  intuitive. It is a good choice in many simulation
  models.
• The shape of a triangular density function is a
  triangle.
• Specified by three easy-to-understand parameters
  the minimum possible value, the most likely
  value, and the maximum possible value.
• Use the _at_RISK function RISKTRIANG to generate
  random numbers.
• Enter the formula RISKTRIANG(MinVal,MLVal,MaxVal)
  in any cell.

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The Poisson and Exponential Distribution
Chapter 12 Waiting line models

• waiting line systems generally contain
  uncertainty.
• Most common probability distributions used to
  model these uncertain quantities are the the
  Poisson distribution (interarrival time) and the
  exponential distribution (service time).

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Example 1 The Exponential and Poisson
Distribution
Chapter 12 Waiting line models

• A bank manager wants to study the congestion at
  the banks automatic teller machines (ATMs).
• During a period of time when business is fairly
  steady, several employees use stopwatches to
  gather data on interarrival times and service
  times.

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Ex. 1 (contd) The Solution
Chapter 12 Waiting line models

• To see whether these times are consistent with
  the exponential distribution, histograms of the
  interarrival times and the service times are
  plotted.

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Important waiting line Relationships
Chapter 12 Waiting line models

• Two general types of outputs that are typically
  calculated in waiting line models are time
  averages and customer averages.
• Typical time averages are
• L expected number of customers in system
• LQ expected number of customers in the waiting
  line
• LS the expected number of customers in service
• P(all idle) probability that all servers are
  idle
• P(all busy) probability that all servers are
  busy

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Chapter 12 Waiting line models
Important waiting line Relationships

• Typical customer averages are
• W expected time spent in the system (waiting or
  being served)
• WQ expected time a customer waits in the
  waiting line
• WS expected time spent in service
• Littles formula relates time averages and
  customer averages.
• ? is the average arrival rate L ?W LQ
  ?WQ LS ?WS

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Chapter 12 Waiting line models

• Two other formulas that relate these quantities.
• All customers are either in service or in the
  waiting line, so LLQLS
• Time spent in the system is the time spent in the
  waiting line plus the time spent in
  service WWQWS
• Server utilization, denoted by U, is the long-run
  fraction of time a typical server is busy.

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Analytical waiting line Models
Chapter 12 Waiting line models

• Basic Single-Server Model
• M/M/1 model Kendalls notation
• First M implies that there is the Poisson
  distribution of interarrival times
• Second M implies that the distribution of service
  times is exponential
• 1 implies that there is a single server
• ? is the average service rate
• ? ?/? is the traffic intensity which is very
  useful for measuring the congestion of the system

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Example 2 Basic Single-Server Model
Chapter 12 Waiting line models

• The Smalltown postal branch employs a single
  clerk.
• Customers arrive at this postal branch according
  to a Poisson process.
• Rate of 30 customers per hour
• Average service time is exponentially distributed
  with mean 1.5 minutes.
• All arriving customers enter the branch,
  regardless of the number already waiting in line.
• The manager would like to decide whether to
  improve the system.
• To do this, she first needs to develop a waiting
  line model that describes the steady-state
  characteristics of the current system.

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Ex. 2 (contd) The Solution
Chapter 12 Waiting line models

• Must choose a common unit of time and then
  express the arrival and service rates (? and ?)
  in this unit.
• Could measure time in seconds, minutes, hours, or
  any other convenient time unit, as long as they
  are consistent.
• Will use minutes. Then, because 1 customer
  arrives every 2 minutes, ? ½. Also, ? 0.667.
• The traffic intensity is ? ?/? or 0.75.
• The system is stable and steady state will occur
  because of this being less than 1.

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Chapter 12 Waiting line models

• In general, the formulas for the M/M/1 model are
  somewhat complex.
• Use the M/M/1 template file.

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Chapter 12 Waiting line models

• Results from completed template
• The arrival rate is 0.5 and the service rate is
  0.667, the expected number of customers in the
  waiting line is 2.25 and the expected time a
  typical customer spends in the waiting line is
  4.5 minutes.
• However 25 of all customers spend no time in the
  waiting line, while 53.7 spend more than 2
  minutes in the waiting line.
• The steady probability of having exactly 4
  customers in the system is 0.079. Equivalently,
  there are exactly 4 customers in the system 7.9
  of the time.
• The bank manager can experiment with other
  arrival rates or services rates in cells B5 and
  B6 to see how the various output measures are
  affected.

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Chapter 12 Waiting line models

• One particularly important insight can be
  obtained through a data table.

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Chapter 12 Waiting line models

• The table shows how bad things can get when the
  service rate is just barely above the arrival
  rate, so that the server utilization is just
  barley below 1.
• The corresponding line chart shows that the
  expected time in waiting line increases extremely
  rapidly as the service rate approaches the
  arrival rate.
• The manager now knows that she does not want a
  services rate close to the arrival rate, at least
  not for extended periods of time.

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Analytical waiting line Models
Chapter 12 Waiting line models

• Basic Multiple-Server Model
• The simplest version of this multiple-server
  parallel system, labeled the M/M/s model
• The s in M/M/s denotes the number of servers.
• There are two types of waiting line
  configurations
• One where each server has a separate line.
  Customers decide which line to join.
• One where there is a single waiting line.
  Customers are served in the FCFS fashion.
• The M/M/s assumes that all customers wait in a
  single line and are served in FCFS fashion.

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Chapter 12 Waiting line models
Analytical waiting line Models
Multiple-server Model

• There are three inputs to this system
• The arrival rate ?
• The service rate (per server) ?
• The number of servers s.
• Assume that the traffic intensity is less than 1.
• Arrival rate ? is less than the maximum service
  rate s?
• The associated formulas are complex. The are
  implemented with a VBA macro in Excel.

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Example Multiple-server Model
Chapter 12 Waiting line models
Analytical waiting line Models

• County Bank has several branch location.
• Customers arrive at a Poisson rate of 150 per
  hour.
• The branch employs 6 tellers.
• Each teller takes, on average, 2 minutes to serve
  a customer, and service times are exponentially
  distributed.
• All tellers perform all tasks, so that customers
  can go to any of the 6 tellers.
• Customers who arrive and find all 6 servers busy
  join a single waiting line and are then served in
  FCFS fashion.
• The manager wants to develop a waiting line model
  of the current system.
• He then wants to find the best number of
  tellers, given that tellers are paid 8 per hour.

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The Solution
Chapter 12 Waiting line models

• An M/M/s template is available

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Chapter 12 Waiting line models

• Enter the inputs in cells B4 through B7 and then
  click on the button to use the template.
• The template file uses a macro to calculate the
  probability that the system is empty. Built-in
  formulas then calculate all other steady-state
  measures.
• It is determined that
• There are 6 tellers and the server utilization is
  0.833.
• The expected number of customers in the system is
  7.94 and the expected time a typical customer
  spends in the system if 0.053 hour.
• The server utilization in an M/M/s system,
  calculated as the arrival rate divided by the
  maximum service rate, is always the expected
  fraction of time a typical server is busy.

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Economic Analysis
Chapter 12 Waiting line models

• There is a cost and benefit from adding a teller.
• The cost is the wage rate paid to the extra
  teller, 8 per hour.
• The benefit is that customers wait less time in
  the bank.
• The problem is evaluating the cost of waiting in
  line.
• Key to the trade-off is assessing a unit cost,
  cQ, per customer per hour of waiting in the
  waiting line.
• If the manager can assess this unit cost, then
  the total expected cost per hour is cQ?WQ.
• Trade off this waiting cost against the cost of
  hiring extra tellers.

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Chapter 12 Waiting line models
The general shape of waiting cost
Total cost
Service cost
Total cost per hour
Waiting cost
Number of channels
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Is the total cost per time period optimal ?
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Chapter 12 Waiting line models
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Analytical waiting line Models
Chapter 12 Waiting line models

• Other Exponential Models
• Limited waiting room model
• Basic model but assumes that arrivals are turned
  away when the number already in the waiting line
  is at some maximum level.
• Limited source model
• Assumes that there are only a finite number of
  customers in the entire population

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Chapter 12 Waiting line models
Analytical waiting line Models

• Erlang Loss Model
• In this model there is no waiting room so
  customers who arrive when all servers are busy
  are lost to the system.
• The steady-state distribution depends on the
  service time distribution only through its mean.
• General Multiple-Server Model
• Variation of M/M/s is to allow nonexponential
  interarrival and/or service times.
• G/G/s allows any interarrival time distribution
  and any service time distribution.
• Difficult to obtain exact analytical results.
• There is an approximation to this model that
  gives quite accurate results Allen-Cunneen
  approximation

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Waiting line Simulation Methods
Chapter 12 Waiting line models

• A popular alternative to analytical models is to
  develop waiting line simulations.
• Advantages
• Not restricted to the assumptions required by the
  standard analytical waiting line models
• Get to see the action through time
• Downside
• Traditionally difficult requiring computer
  programming skills

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Chapter 13 Simulation

• Learning objectives
• Understand what simulation is and how it aids in
  the analysis of a problem.
• Learn why simulation is a significant
  problem-solving tool.
• Understand the difference between static and
  dynamic simulation.
• Identify the important role probability
  distributions, random numbers, and the computer
  play in implementing simulation models.

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Chapter 13 Simulation

• A simulation model is a computer model that
  imitates a real-life situation.
• The fundamental advantage of a simulation model
  is that it shows an entire distribution of
  results, not simply a single bottom-line result.
• Simulation allows us to generate many scenarios,
  each leading to a particular net present value
  (NPV).
• Simulation models are useful for determining how
  sensitive a system is to changes in operating
  conditions.
• Primary difference between analytical models and
  simulation models is that at least one of the
  input variable in a simulation model contains
  random numbers
• Models that simulates how the system changes over
  time are dynamic simulation models

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Chapter 13 Simulation
To simulate means to assume the appearance of
characteristics of reality. In DSSs simulation
generally refers to a technique for conducting
experiments, such as what-if analysis, with a
computer on a model of a management system. Such
type of the model is called a simulation
model. Simulation is one of the most frequently
used tools of DSSs. Reason DSS deals with
semistructured or unstructured situations which
involves complex reality which may not be easily
represented by optimization or other standard
models but often can be handled by simulation.
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Chapter 13 Simulation
Simulation is not a regular type of model. Models
in general represent reality, whereas simulation
usually imitates it closely. Consequence There
are fewer simplifications of reality in
simulation models than in other
models. Simulation can describe or predict
characteristics of a given system under different
circumstances. Consequence Once the
characteristics values are computed, the best
among several alternatives can be selected. The
simulation process often consists of the
repetition of an experiment many times to obtain
an estimate of the overall effect of certain
actions. Consequence A computer is usually
needed.
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Chapter 13 Simulation

• Advantages of simulation
• Simulation is used for decision support because
  it
• Allows for inclusion of the real-life
  complexities of problems. For example, simulation
  may utilize the real life probability
  distributions rather than approximate theoretical
  distributions.
• Is descriptive.This allows the manager to ask
  what-if type questions. Thus, managers who
  employ a trial-and-error approach to problem
  solving can do it faster and cheaper, with less
  risk, using a simulated problem instead of a real
  one.

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Chapter 13 Simulation

• Advantages of simulation
• Simulation is used for decision support because
  it
• Can handle an extremely wide variation in problem
  types, such as inventory and staffing, as well as
  higher managerial-level tasks like long range
  planning.The manager can experiment with
  different variables to determine which are
  important, and with different alternatives to
  determine which is best.
• Can show the effect of compressing time, giving
  the manager in a matter of minutes some