Course Development Team
Head of Programme
: Dr Huong Ha
Head, Business and
: Dr Chang Young Ho
Management Minors
Course Developer(s)
: Dr Zhou Zihan
Technical Writer
: Adeline Loh, ETP
©
2019 Singapore University of Social Sciences. All rights reserved.
No part of this material may be reproduced in any form or by any means without
permission in writing from the Educational Technology & Production, Singapore
University of Social Sciences.
ISBN 9789814810173
Educational Technology & Production
Singapore University of Social Sciences
463 Clementi Road
Singapore 599494
How to cite this Study Guide (APA):
Zhou, Z. (2019). BUS107 Quantitative Methods (Study Guide). Singapore: Singapore
University of Social Sciences.
Release V2.6
Table of Contents
Table of Contents
Course Guide
1. Welcome.................................................................................................................. CG-2
2. Course Description and Aims............................................................................ CG-3
3. Learning Outcomes.............................................................................................. CG-5
4. Learning Material................................................................................................. CG-7
5. Assessment Overview.......................................................................................... CG-8
6. Course Schedule.................................................................................................. CG-10
7. Learning Mode.................................................................................................... CG-11
Study Unit 1: Introduction to Quantitative Analysis and Linear
Programming
Learning Outcomes................................................................................................. SU1-2
Overview................................................................................................................... SU1-4
Chapter 1: Introduction to Quantitative Analysis.............................................. SU1-5
Chapter 2: Linear Programming (LP)................................................................... SU1-9
Summary................................................................................................................. SU1-27
Formative Assessment.......................................................................................... SU1-28
References............................................................................................................... SU1-40
Study Unit 2: Forecasting and Decision Analysis
Learning Outcomes................................................................................................. SU2-2
i
Table of Contents
Overview................................................................................................................... SU2-3
Chapter 1: Forecasting............................................................................................. SU2-4
Chapter 2: Decision Analysis............................................................................... SU2-16
Summary................................................................................................................. SU2-25
Formative Assessment.......................................................................................... SU2-26
References............................................................................................................... SU2-38
Study Unit 3: Simulation and Network Modelling
Learning Outcomes................................................................................................. SU3-2
Overview................................................................................................................... SU3-3
Chapter 1: Simulation Modelling.......................................................................... SU3-4
Chapter 2: Network Modelling............................................................................ SU3-11
Summary................................................................................................................. SU3-18
Formative Assessment.......................................................................................... SU3-19
References............................................................................................................... SU3-34
ii
List of Tables
List of Tables
Table 1.1 ...................................................................................................................... SU1-12
Table 1.2 ...................................................................................................................... SU1-12
Table 2.1 ........................................................................................................................ SU2-8
Table 2.2 ........................................................................................................................ SU2-9
Table 2.3 ...................................................................................................................... SU2-11
Table 3.1 ........................................................................................................................ SU3-6
Table 3.2 ........................................................................................................................ SU3-7
Table 3.3 ........................................................................................................................ SU3-8
Table 3.4 ........................................................................................................................ SU3-9
iii
List of Tables
iv
List of Figures
List of Figures
Figure 1.1 Layout........................................................................................................ SU1-15
Figure 1.2 Solver Function........................................................................................ SU1-16
Figure 1.3 Adding Constraints................................................................................. SU1-17
Figure 1.4 Final Setup................................................................................................ SU1-17
Figure 1.5 Solver Output Selection.......................................................................... SU1-18
Figure 1.6 Solver Output........................................................................................... SU1-19
Figure 1.7 Sensitivity Analysis................................................................................. SU1-23
Figure 3.1 Network of Expressways........................................................................ SU3-17
v
List of Figures
vi
List of Lesson Recordings
List of Lesson Recordings
Introduction to Linear Programming....................................................................... SU1-9
Introduction to Linear Programming – Model Structure..................................... SU1-14
Introduction to Linear Programming – Model Formulation................................ SU1-14
Introduction to Linear Programming – Solving Linear Programming Model
Graphically................................................................................................................... SU1-15
Linear Programming Sensitivity Analysis – Basics............................................... SU1-22
Linear Programming Sensitivity Analysis – Range of Optimality...................... SU1-24
Linear Programming Sensitivity Analysis – Range of Optimality & 100%
Rule................................................................................................................................ SU1-24
Linear Programming Sensitivity Analysis – Range of Feasibility (Dual
Price).............................................................................................................................. SU1-25
Time Series & Smoothing Methods in Forecasting – Components of Time
Series............................................................................................................................... SU2-5
Time Series & Smoothing Methods in Forecasting – Moving Average & Centered
Moving Average............................................................................................................ SU2-7
Time Series & Smoothing Methods in Forecasting – Exponential
Smoothing...................................................................................................................... SU2-8
Time Series & Smoothing Methods in Forecasting – Trend Projection............... SU2-13
Time Series & Smoothing Methods in Forecasting – Seasonal
Components................................................................................................................. SU2-14
vii
List of Lesson Recordings
Time Series & Smoothing Methods in Forecasting – Trend and Seasonal
Components................................................................................................................. SU2-14
Problem Formulation and Decision Making Without Probabilities – Payoff
Table.............................................................................................................................. SU2-17
Problem Formulation and Decision Making Without Probabilities – Optimistic
Approach...................................................................................................................... SU2-18
Problem Formulation and Decision Making Without Probabilities –
Conservative Approach............................................................................................. SU2-18
Problem Formulation and Decision Making Without Probabilities – MiniMax
Regret Approach......................................................................................................... SU2-19
Decision Making With Probabilities – Decision Tree............................................ SU2-21
Decision Making With Probabilities – Expected Value Approach and Expected
Value of Perfect Information..................................................................................... SU2-22
Decision Making With Probabilities – Risk Profile................................................ SU2-23
Simulation Modelling and Applications – Purpose of Simulation........................ SU3-5
Simulation Modelling and Applications – Monte Carlo Simulation..................... SU3-6
Simulation Modelling and Applications – Monte Carlo Simulation using
Probabilities Interval.................................................................................................... SU3-6
Network Models – Shortest Route........................................................................... SU3-14
Network Models – Maximum Flow........................................................................ SU3-15
Network Models – Minimum Spanning Tree......................................................... SU3-16
viii
Course
Guide
Quantitative Methods
BUS107
Course Guide
1. Welcome
Presenter: Dr Yuan Xuchuan
This streaming video requires Internet connection.
Access it via Wi-Fi to avoid incurring data charges on your personal mobile plan.
Click here to watch the video. i
Welcome to your study of BUS107 Quantitative Methods, a 5 credit unit (CU) course.
This Study Guide is divided into two sections – the Course Guide and Study Units.
The e-Course Guide provides a structure for the entire course. As the phrase implies, the
e-Course Guide aims to guide you through the learning experience. In other words, it may
be seen as a roadmap through which you are introduced to the different topics within
the broader subject. This Guide has been prepared to help you understand the learning
objectives of the course. In addition, it explains how the various materials and resources
are organized, and how they may be used; how your learning will be assessed; and how
to get help if you need it.
i
https://d2jifwt31jjehd.cloudfront.net/BUS107/IntroVideo/BUS107_Intro_Video.mp4
CG-2
BUS107
Course Guide
2. Course Description and Aims
BUS107 Quantitative Methods introduces the essential concepts of quantitative methods
that are commonly practiced in business and management for decision-making and
resource planning purposes. It examines a series of quantitative techniques that are of
interest and relevance to practitioners and researchers in this field. The underlying theme
behind each quantitative technique is the formulation of an appropriate quantitative
model. Students studying this course will learn the technique of quantitative model
formulation and processing, using relevant computer software to solve practical business
problems. The critical skills of data analysis and interpretation for decision making will
also be taught in this course. Students will learn to work in teams to solve cases as well as
present the findings in class.
Course Structure
This course is a 5-credit unit course presented over 6 weeks.
There are three Study Units in this course. The following provides an overview of each
Study Unit.
Study Unit 1 – Introduction to Quantitative Analysis and Linear Programming
This unit provides an overview of quantitative analysis, and to explain the reason for its
study. Develop a general understanding of the quantitative analysis approach to decision
making and realise that quantitative applications begin with a problem situation. Linear
Programming and its applications is the main focus of this unit.
Study Unit 2 – Forecasting and Decision Analysis
This unit aims to provide an understanding of the modeling techniques used in forecasting
as well as to show how decision analysis can help organizations make effective decisions
when faced with uncertainty circumstances.
CG-3
BUS107
Course Guide
Study Unit 3 – Simulation and Network Modelling
The aim of this unit is to introduce the modeling techniques and applications of simulation
to a variety of situations in the analysis of problems. This unit also discuss the concepts and
applications of network modeling, covering popular network models that solve specific
needs.
CG-4
BUS107
Course Guide
3. Learning Outcomes
Knowledge & Understanding (Theory Component)
By the end of this course, you should be able to:
• Describe the management science/operations research approach to decision
making.
• Apply linear programming models for simple problems.
• Interpret the solution of a linear programming problem for business decisionmaking.
• Employ the techniques of classical time series modelling.
• Use classical time series modeling to predict future aspects of business operations.
• Discuss a simple decision analysis problem from both a payoff table and decision
tree point of view as to develop a risk profile and interpret its meaning for business
decision-making.
• Define what simulation is and explain how it aids in the analysis of a problem.
• Develop network and linear programming models for the minimal-spanning tree,
the maximum-flow and the shortest-route problems.
Key Skills (Practical Component)
By the end of this course, you should be able to:
• Use suitable computer software to construct and process quantitative models for
result generation and reporting.
• Identify alternatives to decision-making problems through data analysis and
interpretation of the results derived from the quantitative model.
• Develop decision alternatives in a logical and concise manner.
• Develop the essential knowledge and interpersonal skills to work effectively in a
team.
CG-5
BUS107
Course Guide
• Illustrate the results of various areas related to Quantitative Methods in class.
CG-6
BUS107
Course Guide
4. Learning Material
The following is a list of the required learning materials to complete this course.
Required Textbook(s)
Winston, W. L., & Albright, S. C. (2019). Practical management science (Sixth ed.).
Cengage Learning. Retrieved from https://online.vitalsource.com/#/
books/9789814844925/cfi/0!/4/2@100:0.00
To launch eTextbook, you need a VitalSource account which can be created via Canvas
(iBookstore), using your SUSS email address. Access to this eTextbook is restricted by
enrolment to this course.
Other recommended study material
1
Software
Microsoft Office (with Excel Solver)
CG-7
BUS107
Course Guide
5. Assessment Overview
The overall assessment weighting for this course for the Evening Cohort is as follows:
Assessment
Description
Weight
Allocation
Pre-Course Quiz 01
2%
Pre-Class Quiz 01
2%
Pre-Class Quiz 02
2%
Assignment 2
Group-based Assignment 1
38%
Class Participation
Participation during seminars
6%
Exam
Written Examination
50%
Assignment 1
TOTAL
100%
The overall assessment weighting for this course for the Day-time Cohort is as follows:
Assessment
Assignment 1
Assignment 2
Description
Weight
Allocation
Pre-Course Quiz 01
2%
Pre-Course Quiz 02
2%
Pre-Course Quiz 03
2%
Group-based Assignment 1
38%
CG-8
BUS107
Course Guide
Assessment
Description
Weight
Allocation
Class Participation
Participation during seminars
6%
Exam
Written Examination
50%
TOTAL
100%
SUSS’s assessment strategy consists of two components, Overall Continuous Assessment
Scores (OCAS) and Overall Examinable Scores (OES) that make up the overall course
assessment score.
For SBIZ courses, both components will be equally weighted: 50% OCAS and 50% OES.
a.
OCAS: In total, this continuous assessment will constitute 50 percent of
overall student assessment for this course. The continuous assignments are
compulsory and are non-substitutable. It is imperative that you read through
your Assignment questions and submission instructions before embarking on
your Assignment.
b.
OES: The Examination is 100% of this component.
To be sure of a pass result you need to achieve scores of 40% in each component.
Your overall rank score is the weighted average of both components.
CG-9
BUS107
Course Guide
6. Course Schedule
Flexible learning – learning at your own pace, space and time -- is a hallmark at SUSS. One
of your greatest challenges is to manage your time well so you can meet given deadlines.
To help keep your study progress in check, you should pay special attention to your
Course Schedule. It contains study unit related activities including Assignment, selfevaluations, and examinations. Please refer to the Course Timetable in the Student Portal
for the updated Course Schedule.
NOTE: You should always make it a point to check the Student Portal for any
announcements and latest updates.
You need to ensure you fully understand the contents of each Study Unit listed in the
Course Schedule. It will guide you through activities you are expected to complete either
independently and/or in groups. It will also indicate if/when there is a need for face-toface activities.
The Course Schedule will indicate the number of required Assignments. It is important
you comprehend the Overall Assessment Weighting of your course. This is listed in
Section 5 of this Guide.
Self-evaluations are built into your online learning activities to help you revise and retain
the knowledge garnered, to better prepare you for any required formal assessment. If your
course requires an end-of-semester examination, do look through the Specimen Exam
Paper which is available on Learning Management System.
CG-10
BUS107
Course Guide
7. Learning Mode
The learning process for this course is structured along the following lines of learning:
a.
Self-study guided by the study guide units. Independent study will require at
least 3 hours per week.
b.
Working on assignments, either individually or in groups.
c.
Classroom Seminar sessions (3 hours each session, 3 sessions in total for evening
cohort and 6 sessions in total for daytime cohort).
iStudyGuide
You may be viewing the iStudyGuide version, which is the mobile version of the
Study Guide. The iStudyGuide is developed to enhance your learning experience with
interactive learning activities and engaging multimedia. Depending on the reader you are
using to view the iStudyGuide, you will be able to personalise your learning with digital
bookmarks, note-taking and highlight sections of the guide.
Interaction with Instructor and Fellow Students
Although flexible learning – learning at your own pace, space and time – is a hallmark
at SUSS, you are encouraged to engage your instructor and fellow students in online
discussion forums. Sharing of ideas through meaningful debates will help broaden your
learning and crystallise your thinking.
Academic Integrity
As a student of SUSS, it is expected that you adhere to the academic standards stipulated
in The Student Handbook, which contains important information regarding academic
policies, academic integrity and course administration. It is necessary that you read and
understand the information stipulated in the Student Handbook, prior to embarking on
the course.
CG-11
BUS107
Course Guide
CG-12
Study
Unit
1
Introduction to Quantitative
Analysis and Linear Programming
BUS107
Introduction to Quantitative Analysis and Linear Programming
Learning Outcomes
By the end of this unit, you should be able to:
1.
Develop a general understanding of the quantitative analysis approach to
decision making and realise that quantitative applications begin with a problem
situation.
2.
Identify the step-by-step procedure that is used in most quantitative approaches
to decision making.
3.
Recall basic models of cost, revenue, and profit and be able to compute the breakeven point.
4.
Discuss microcomputer software packages and their role in quantitative
approaches to decision making.
5.
Recognise possible problems in using quantitative analysis.
6.
List the types of problems that linear programming is able to solve.
7.
Develop linear programming models for simple problems.
8.
Identify the special features of a model that make it a linear programming model.
9.
Solve graphically any linear programming (LP) problem having two variables
by using extreme points and the objective function line in obtaining the optimal
solution.
10.
Interpret and use slack and surplus variables.
11.
Use the appropriate software to find optimum solutions.
12.
Explain how alternative optimal solutions, infeasibility, unboundedness and
redundancy can occur in LP problems.
13.
Employ graphical sensitivity analysis for LP problems in two variables.
14.
Interpret the range of optimality for objective function coefficients.
15.
Interpret the dual price for a constraint.
16.
Use computer software packages to formulate, solve and interpret the solution
for linear programmes with more than two decision variables.
SU1-2
BUS107
Introduction to Quantitative Analysis and Linear Programming
SU1-3
BUS107
Introduction to Quantitative Analysis and Linear Programming
Overview
In the quantitative analysis process, it uses a scientific approach to managerial decision
making. This domain of knowledge is also known as Management Science. The approach
to quantitative analysis begins with information and data. The raw data are manipulated
into useful information, and results are then used to make the best decision out of the
number of alternatives.
Chapter 1 describes the quantitative analysis process. It describes how quantitative
methods can be used to solve business problems.
Chapter 2 describes the LP process. It introduces LP, one of the most powerful and flexible
methods of quantitative analysis. Then it examines the sensitivity analysis for LP.
SU1-4
BUS107
Introduction to Quantitative Analysis and Linear Programming
Chapter 1: Introduction to Quantitative Analysis
A mathematical model is a quantitative representation, or idealisation, of a real problem.
It is a key to virtually every management science application. It can be phrased in terms
of mathematical expressions (equations and inequalities) or a series of interrelated cells in
a spreadsheet.
Read
You should now read Winston and Albright (2019), pp.1-18.
1.1 Introduction to Quantitative Analysis
1.1.1 Models
The purpose of a mathematical model is to represent the essence of a problem in a concise
form, providing several advantages:
• Enables managers to understand the problem better;
• Helps to define the scope of the problem, the possible solutions, and the data
requirements;
• Allows analysts to employ a variety of the mathematical solution procedures that
have been developed over the last 50 years;
• The modelling process itself, if done correctly, often helps to “sell” the solution to
the people who must work with the system that is eventually implemented.
Descriptive models: models that simply describe a situation.
Optimisation models: models that suggest a desirable course of action.
Example:
SU1-5
BUS107
Introduction to Quantitative Analysis and Linear Programming
• Waiting line: Convenience store with a single cash register.
• The manager suspects that excessive waiting times in lines to the register hurt the
business.
• The manager builds a mathematical model to help understand the problem, and
suggests improvements to the current situation.
1.1.2 Dealing with Uncertainty
Optimisation models can be classified as deterministic, meaning that there is no
uncertainty about any of the model inputs. Thus, this takes us to simulation models.
Simulation allows the user to see how an output varies, for any given set of decisions, as
uncertain inputs vary over their ranges of possible values.
1.1.3 The Seven-step Modelling Process
Modelling is a process in which one abstracts the essence of a real problem into a model,
spreadsheet or others. The following are the recommended approaches in the modelling
process:
• Step 1: Problem definition
◦ The analyst first defines the organisation’s problem.
◦ Defining the problem includes specifying the organisation’s objectives and
the parts of the organisation that must be studied before the problem can be
solved.
• Step 2: Data collection
◦ After defining the problem, the analyst collects data to estimate the value of
parameters that affect the organisation’s problem.
• Step 3: Model development
◦ In the third step, the analyst develops a model of the problem.
SU1-6
BUS107
Introduction to Quantitative Analysis and Linear Programming
◦ Some of these are deterministic optimisation models, where all of the
problem inputs are assumed to be known and the goal is to determine values
of decision variables that maximise or minimise the objective of the model.
◦ Others are simulation models, where some of the inputs are modelled
with probability distributions. Occasionally, the models are so complex
mathematically that no simple formulae can be used to relate inputs to
outputs.
• Step 4: Model verification
◦ The analyst now tries to determine whether the model developed in the
previous step is an accurate representation of reality.
◦ The model must pass “plausibility checks.” In this case, various input values
and decision variable values are entered into the model to see whether the
resulting outputs are plausible.
• Step 5: Optimisation 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 organisation’s objectives.
◦ Many optimisation models exist, and they will be discussed throughout the
course
• Step 6: Model communication to management
◦ The analyst presents the model and the recommendations from the previous
steps to the organisation.
• Step 7: Model implementation
◦ If the organisation 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 organisation to meet its objectives.
SU1-7
BUS107
Introduction to Quantitative Analysis and Linear Programming
Activity 1
Discuss the difference between modelling and models in the context of management
science.
Review Questions
1.
Identify the type of model that is key to virtually every management science
application. What is the major difference between cross-tabulation and frequency
distribution?
2.
What are the advantages of mathematical models?
3.
What are the properties of a good model?
4.
Identify the desired conditions for a successful model implementation.
SU1-8
BUS107
Introduction to Quantitative Analysis and Linear Programming
Chapter 2: Linear Programming (LP)
This chapter introduces spreadsheet optimisation, one of the most powerful and flexible
methods of quantitative analysis. The specific type of optimisation discussed here is linear
programming (LP). This chapter will introduce the basic elements of LP.
Read
You should now read Winston and Albright (2019), pp.71-77.
Lesson Recording
Introduction to Linear Programming
2.1 Introduction to Linear Programming (LP)
LP is used in many organisations, often on a daily basis, to solve a variety of problems:
• Labour scheduling,
• Inventory management,
• Selection of advertising media,
• Bond trading, etc.
2.1.1 Optimisation
All optimisation models have several common elements:
• Decision variables, whose values the decision maker is allowed to choose. The
values of these variables determine such outputs as total cost, revenue, and profit.
SU1-9
BUS107
Introduction to Quantitative Analysis and Linear Programming
• An objective function (objective, for short) to be optimised—minimised or
maximised.
• Constraints that must be satisfied. They are usually physical, logical, or economic
restrictions, depending on the nature of the problem.
o In searching for the values of the decision variables that optimise the objective,
only those values that satisfy all of the constraints are allowed.
Excel uses its own terminology for optimisation:
• Excel refers to the decision variables as the changing cells. These cells must contain
numbers that are allowed to change freely; they are not allowed to contain formulae.
• Excel refers to the objective as the objective cell.
◦ There can be only one objective cell, which could contain profit, total cost,
total distance travelled, or others, and it must be related through formulae
to the changing cells.
◦ When the changing cells change, the objective cell should change
accordingly.
• Appropriate cells and cell formulae operationalise the constraints, which can come
in a variety of forms.
◦ Non-negativity constraint is very common. It states that changing cells
must have nonnegative (zero or positive) values. Non-negativity constraints
are usually included for physical reasons. For example, it is impossible to
produce a negative number of automobiles.
2.1.2 Steps in Solving an Optimisation Problem
There are two basic steps in solving an optimisation problem:
• Model development step
• Optimisation step
SU1-10
BUS107
Introduction to Quantitative Analysis and Linear Programming
Model development step
In the model development step, you decide what the decision variables are, which
constraints are required, and how everything fits together.
• If you are developing an algebraic model, you must derive correct algebraic
expressions.
• If you are developing a spreadsheet model, you must relate all variables with
appropriate cell formulae.
Optimisation step
To optimise means that you must systematically choose the values of the decision variables
that make the objective as large (for maximisation) or small (for minimisation) as possible
and cause all of the constraints to be satisfied.
Any set of values of the decision variables that satisfies all of the constraints is called a
feasible solution.
The set of all feasible solutions is called the feasible region.
The desired feasible solution is the one that provides the best value – minimum for a
minimisation problem, maximum for a maximisation problem – for the objective. This
solution is called the optimal solution.
Much of the published research has been about the optimisation step. One algorithm for
searching through the feasible region is called the simplex method, programmed into
Excel’s Solver add-in.
2.1.3 Worked Example
Supposed a manufacturer makes two types of toy cars, Speedy and Zippy. The following
table shows the resource hours needed and the hours available each week:
SU1-11
BUS107
Introduction to Quantitative Analysis and Linear Programming
Table 1.1
Type of Toy Cars
Hours of
Moulding
Hours of
Trimming
Speedy
Zippy
Available
2
3
19
1
0
6
They have limited supply of plastic. The amounts needed for each type of plastic and total
amount available each week, are given below:
Table 1.2
Type of Plastic
Speedy
Zippy
Available
Steely
1
1
8
The net profit on each batch of toy cars is $5 for Speedy and $7 for Zippy.
Formulate this as a linear programme.
Step 1: Define the Decision Variables
Since the whole case revolves around the decisions to be made on the quantity of Speedy
and Zippy toy cars to be produced, hence, we note that there are two decision variables
that need to be defined.
Let
= the amount of Speedy toy cars to produce
= the amount of Zippy toy cars to produce
Step 2: Define the Type of Optimisation Problem
As the profit margin for each car type is given, the case is concerned with the best
profit to be made in view of the constraints in raw materials. Hence, this is a case of
SU1-12
BUS107
Introduction to Quantitative Analysis and Linear Programming
maximisation problem, where the purpose is to maximise the profit through the best
possible combination of quantity of Speedy and Zippy toy cars to be produced.
Step 3: Define the Objective Function
Since step 2 suggested that this case is a maximisation problem, and it is about maximising
the profit, hence, the objective function would naturally be the equation that represents
total profit.
Profit = Z = 5
+7
Step 4: Determine the Constraints
Constraints typically represent either the capacity/resource limit (e.g., required capacity
≤ available capacity) or the commitment required to be fulfilled (e.g., produced quantity
≥ customer demand).
In this case, there are 3 constraints and they are due to limited resources. For example, on
the Moulding constraint:
Since 1 unit of Speedy toy car would need 2 units of Moulding, hence,
car would need 2
unit of Speedy toy
units of Moulding. Likewise, since 1 unit of Zippy toy car would need
3 units of Moulding, hence,
unit of Speedy toy car would need 3
Thus, the total consumption for Moulding is: 2
+3
units of Moulding.
and we know that the available
supply of Moulding hours is 19. Hence, this constraint can be written as:
Constraint 1:
2
+3
≤
19
Moulding
Following the same logic, we can write the constraints for Trimming and Steely:
Constraint 2:
1
+0
≤
6
Trimming
Constraint 3
1
+1
≤
8
Steely
Last but not least, since we are dealing with decision variables that cannot take on
negative values, hence, to ensure that the model reflects this information, the nonnegativity constraints need to be included for every LP model. Thus,
SU1-13
BUS107
Introduction to Quantitative Analysis and Linear Programming
Non-
,
negativity:
≥
0
Step 5: Complete Model:
Let
= Production for Speedy
and
= Production for Zippy
Max. 5
+7
Subject to
2
+3
≤
19
Moulding
1
+0
≤
6
Trimming
1
+1
≤
8
Steely
≥
0
,
Lesson Recording
Introduction to Linear Programming – Model Structure
Introduction to Linear Programming – Model Formulation
2.1.4 Solving LP Model
There are several ways of solving the LP Model:
• Graphical
• Software
SU1-14
BUS107
Introduction to Quantitative Analysis and Linear Programming
Lesson Recording
Introduction to Linear Programming – Solving Linear Programming Model
Graphically
Here, we are going to use Microsoft Excel to solve. First, the model is written in this layout,
where the coloured cells (C6 and D6) represent the final decision to be made.
Figure 1.1 Layout
These are the procedures:
1.
Activate the Solver function: Data>Solver.
SU1-15
BUS107
Introduction to Quantitative Analysis and Linear Programming
2.
Choose Cell C8 as the cell to be optimised, in particular, checked on Max.
3.
Indicate the location of the decision variables; namely C6 and C8 in the field ‘By
Changing Variable Cells’.
4.
Click on ‘Add’ so that constraint information could be added.
Figure 1.2 Solver Function
SU1-16
BUS107
Introduction to Quantitative Analysis and Linear Programming
Figure 1.3 Adding Constraints
Figure 1.4 Final Setup
SU1-17
BUS107
Introduction to Quantitative Analysis and Linear Programming
Figure 1.5 Solver Output Selection
SU1-18
BUS107
Introduction to Quantitative Analysis and Linear Programming
Figure 1.6 Solver Output
The Solver’s recommended solution:
To produce 5 Speedy toy cars and 3 Zippy toy cars which would potentially yield the best
possible profit of $46.
SU1-19
BUS107
Introduction to Quantitative Analysis and Linear Programming
2.1.5 Assumptions and Limitations of LP
The key assumption for LP technique is that the constraints and the objective function
are linear. Also, there are three other important properties that LP models possess that
distinguish them from general mathematical programming models:
• proportionality,
• additivity,
• and divisibility.
Any violations of the above properties would limit the use of LP technique.
Read
You should now read Winston and Albright (2019), pp.97-100.
2.1.6 Alternative Optimisation Models
In situations where the decision variables can only be integer, i.e., it does not make sense
for them to be divisible, an extension of LP technique would be needed. This is called
Integer Programming.
Read
You should now read Winston and Albright (2019), pp.278-324.
In many complex optimization problems, the objective and/or the constraints are
nonlinear functions of the decision variables. Such optimization problems are called
Nonlinear Programming (NLP) problems or Nonlinear Optimisation models.
SU1-20
BUS107
Introduction to Quantitative Analysis and Linear Programming
Read
You should now read Winston and Albright (2019), pp.340-398.
2.1.7 Infeasibility and Unboundedness
An infeasible solution is a solution that violates at least one constraint. Infeasible solutions
are disallowed. Infeasibility occurs when there are no feasible solutions to the model.
There are generally two reasons:
• There is a mistake in the model (an input was entered incorrectly, such as a ≤ symbol
instead of a ≥), or
• The problem has been so constrained that there are no solutions left.
The optimum objective value is said to be unbounded if it can be made as large (or as
small, for minimisation problem) as you like.
• If this occurs, you have probably entered a wrong input or forgotten some
constraints.
Read
You should now read Winston and Albright (2019), pp.100-101.
2.2 Linear Programming (LP) Sensitivity Analysis
Solver offers you the option to obtain a sensitivity report.
SU1-21
BUS107
Introduction to Quantitative Analysis and Linear Programming
• The report is based on a well-established theory of sensitivity analysis in
optimisation models.
• Solver’s sensitivity report performs two types of sensitivity analysis:
1.
On the coefficients of the objectives, (i.e.,range of optimality), and
2.
On the right-hand sides of the constraints, (i.e.,range of feasibility).
Read
You should now read Winston and Albright (2019), pp.87-97.
Lesson Recording
Linear Programming Sensitivity Analysis – Basics
2.2.1 Sensitivity Analysis Report
Using the same example in section 2.1.3, the following is the sensitivity analysis report:
SU1-22
BUS107
Introduction to Quantitative Analysis and Linear Programming
Figure 1.7 Sensitivity Analysis
2.2.2 Range of Optimality
In Figure 1.7, row 6 to row 10 show the Range of Optimality output. It simply states that
as long as the objective function coefficient increases or decreases within the allowable
range, keeping the other objective function coefficient constant, the optimal value for the
decision variables remains unchanged.
However, if both objective function coefficients change simultaneously, we would need to
use the 100% Rule to determine if there is a change in solution. The 100% rule states that
simultaneously changing the Objective Function coefficients will not change the Optimal
SU1-23
BUS107
Introduction to Quantitative Analysis and Linear Programming
Solution if the sum of the percentages of the change divided by the maximum allowable
change for each coefficient does not exceed 100%. However, the 100% rule does NOT say
that the optimal solution will change if the sum of the percentage changes exceeds 100%.
For Example:
C1, Objective Coefficient for
= 6,
C2, Objective Coefficient for
= 7.2
Change:
Max allowable increase / decrease:
C1
C2
from 5 to 6 = 1
from 7 to 7.2 = 0.2
2
0.5
1/2 = 50%
0.2/0.5 = 40%
(same direction as changes)
Percentage change
Sum of percentage changes
50% + 40% = 90%
Since 90% < 100%, therefore the 100% Rule is satisfied; simultaneously changing both C1
& C2 does not affect the optimal solution point (as indicated by the Range of Optimality).
Lesson Recording
Linear Programming Sensitivity Analysis – Range of Optimality
Linear Programming Sensitivity Analysis – Range of Optimality & 100% Rule
SU1-24
BUS107
Introduction to Quantitative Analysis and Linear Programming
2.2.3 Range of Feasibility
On the range of feasibility, the concern is whether by changing the RHS values, it does
affect the dual price (also called shadow price).
For example, Constraint 1 in Figure 1.7 indicates a dual price of 2, which is valid if the
RHS ranges from 18 to 24, while holding the other constraint’s RHS unchanged. The 100%
Rule can be used for concurrent changes in the RHS values.
For each constraint, there is a dual price associated with it. The dual price gives the change
in the optimal objective value if the RHS of the constraint changes by one unit. In the
above example, if the available hours of moulding increases from 19 to 20, total profit will
increase by $2. Note that when a constraint is not binding, its dual price is 0.
Lesson Recording
Linear Programming Sensitivity Analysis – Range of Feasibility (Dual Price)
Activity 2
Discuss the difference between range of optimality and the range of feasibility in the
sensitivity analysis of LP model.
Review Questions
1.
If a manufacturing process takes 4 hours per unit of x and 2 hours per unit of y and
a maximum of 100 hours of manufacturing process time is available, formulate the
algebraic formulation of this constraint.
SU1-25
BUS107
2.
Introduction to Quantitative Analysis and Linear Programming
Suppose a company sells two different products, x and y, for net profits of $6 per unit
and $3 per unit, respectively. What is the slope of the line representing the objective
function?
3.
What are the three important properties in LP models?
4.
Consider the following LP problem:
Maximise 4
+2
Subject to:
4
+2
2
+
,
≥ 40
≥ 20
≥0
Identify the above LP problem.
SU1-26
BUS107
Introduction to Quantitative Analysis and Linear Programming
Summary
This chapter has provided background to LP modelling and to optimisation modelling in
general.
This chapter also introduces ways to develop basic LP spreadsheet models, and how to
use Solver to find the optimal solutions, and how to perform sensitivity analyses with
Solver’s sensitivity reports.
SU1-27
BUS107
Introduction to Quantitative Analysis and Linear Programming
Formative Assessment
1.
Which of the following is a type of model that is key to virtually every management
science application?
a. Heuristic model
b. Queuing model
c. Mathematical model
d. Regression model
2.
Before trusting the answers to what-if scenarios from a spreadsheet model, a manager
should attempt to:
a. validate the model
b. make sure all possible scenarios have been investigated
c. check the mathematics in the model
d. review the model
3.
Optimization models are useful for determining:
a. sensitivity to inputs
b. whether the inputs are valid or not
c. what the manager should do
d. the value of the output under the current conditions
4.
Defining an organization's problem includes:
a. specifying the organization's objectives
b. collecting the organization's historical data
c. defining the model of the problem
d. sensitivity analysis
5.
Which of the following is not necessarily a property of a good model?
SU1-28
BUS107
Introduction to Quantitative Analysis and Linear Programming
a. The model represents the client's real problem accurately
b. The model is as simple as possible
c. The model is based on a well-known algorithm
d. The model is the one that the client can understand
6.
The condition of nonnegativity requires that:
a. the objective function cannot be less than zero
b. the decision variables cannot be less than zero
c. the right-hand side of the constraints cannot be greater than zero
d. the reduced cost cannot be less than zero
7.
If a manufacturing process takes 4 hours per unit of x and 2 hours per unit of y and a
maximum of 100 hours of manufacturing process time is available, then an algebraic
formulation of this constraint is:
a. 4x + 2y ≥ 100
b. 4x − 2y ≤ 100
c. 4x + 2y ≤ 100
d. 4x − 2y ≥ 100
8.
The feasible region in all linear programming problems is bounded by:
a. corner points
b. hyperplanes
c. an objective line
d. all of these options
9.
Suppose a company sells two different products, x and y, for net profits of $6 per unit
and $3 per unit, respectively. The slope of the line representing the objective function
is:
a. 0.5
b. −0.5
SU1-29
BUS107
Introduction to Quantitative Analysis and Linear Programming
c. 2
d. −2
10. The equation of the line representing the constraint 4x + 2y ≤ 100 passes through the
points:
a. (25,0) and (0,50)
b. (0,25) and (50,0)
c. (−25,0) and (0,−50)
d. (0,−25) and (−50,0)
11. When the profit increases with a unit increase in a resource, this change in profit will
be shown in the Solver's sensitivity report as the:
a. add-in price
b. sensitivity price
c. shadow price
d. additional profit
12. Linear programming models have three important properties. They are:
a. optimality, additivity and sensitivity
b. optimality, linearity and divisibility
c. divisibility, linearity and nonnegativity
d. proportionality, additivity and divisibility
13. Consider the following linear programming problem:
Maximize 4
+2
Subject to:
4
+2
2
+
,
≤ 40
≥ 20
≥0
SU1-30
BUS107
Introduction to Quantitative Analysis and Linear Programming
The above linear programming problem:
a. has only one feasible solution
b. has more than one optimal solution
c. exhibits infeasibility
d. exhibits unboundedness
14. In a linear programming model, if the constraint’s Right-Hand-Side value exceeds
the range of feasibility, then:
a. the dual price will remain the same.
b. the dual price might change.
c. the constraint will be unbounded.
d. there will be no feasible solution.
SU1-31
BUS107
Introduction to Quantitative Analysis and Linear Programming
Solutions or Suggested Answers
Chapter 1 Review Questions
1.
Identify the type of model that is key to virtually every management science
application. What is the major difference between cross-tabulation and frequency
distribution?
Mathematical model.
2.
What are the advantages of mathematical models?
These are the advantages of mathematical models:
• Mathematical models enable managers to understand the problem better.
• Mathematical models allow analysts to employ a variety of mathematical
solution procedures.
• The mathematical modelling process itself, if done correctly, often helps
"sell" the solution.
3.
What are the properties of a good model?
A good model should have the following properties:
• The model represents the client's real problem accurately.
• The model is as simple as possible.
• The model is one the client can understand.
4.
Identify the desired conditions for a successful model implementation.
These are the desired conditions:
• The people who will run the model understand how to enter appropriate
inputs.
SU1-32
BUS107
Introduction to Quantitative Analysis and Linear Programming
• The people who will run the model are able to run what-if analysis.
• The people who will run the model are able to interpret the model's outputs
correctly.
Chapter 2 Review Questions
1.
If a manufacturing process takes 4 hours per unit of x and 2 hours per unit of y and
a maximum of 100 hours of manufacturing process time is available, formulate the
algebraic formulation of this constraint.
4x + 2y ≤ 100
2.
Suppose a company sells two different products, x and y, for net profits of $6 per unit
and $3 per unit, respectively. What is the slope of the line representing the objective
function?
–2
3.
What are the three important properties in LP models?
Proportionality, additivity and divisibility
4.
Consider the following LP problem:
Maximise 4
+2
Subject to:
4
+2
2
+
,
≥ 40
≥ 20
≥0
Identify the above LP problem.
Unbounded solution and redundant constraint
SU1-33
BUS107
Introduction to Quantitative Analysis and Linear Programming
Formative Assessment
1.
Which of the following is a type of model that is key to virtually every management
science application?
a.
Heuristic model
Incorrect. Refer to Study Unit 1, Chapter 1.
b.
Queuing model
Incorrect. Refer to Study Unit 1, Chapter 1.
c.
Mathematical model
Correct.
d.
Regression model
Incorrect. Refer to Study Unit 1, Chapter 1.
2.
Before trusting the answers to what-if scenarios from a spreadsheet model, a manager
should attempt to:
a.
validate the model
Correct.
b.
make sure all possible scenarios have been investigated
Incorrect. Refer to Study Unit 1, Chapter 1
c.
check the mathematics in the model
Incorrect. Refer to Study Unit 1, Chapter 1
d.
review the model
Incorrect. Refer to Study Unit 1, Chapter 1
3.
Optimization models are useful for determining:
a.
sensitivity to inputs
Incorrect. Refer to Study Unit 1, Chapter 1
SU1-34
BUS107
Introduction to Quantitative Analysis and Linear Programming
b.
whether the inputs are valid or not
Incorrect. Refer to Study Unit 1, Chapter 1
c.
what the manager should do
Correct.
d.
the value of the output under the current conditions
Incorrect. Refer to Study Unit 1, Chapter 1
4.
Defining an organization's problem includes:
a.
specifying the organization's objectives
Correct.
b.
collecting the organization's historical data
Incorrect. Refer to Study Unit 1, Chapter 1
c.
defining the model of the problem
Incorrect. Refer to Study Unit 1, Chapter 1
d.
sensitivity analysis
Incorrect. Refer to Study Unit 1, Chapter 1
5.
Which of the following is not necessarily a property of a good model?
a.
The model represents the client's real problem accurately
Incorrect. Refer to Study Unit 1, Chapter 1
b.
The model is as simple as possible
Incorrect. Refer to Study Unit 1, Chapter 1
c.
The model is based on a well-known algorithm
Correct.
d.
The model is the one that the client can understand
SU1-35
BUS107
Introduction to Quantitative Analysis and Linear Programming
Incorrect. Refer to Study Unit 1, Chapter 1
6.
The condition of nonnegativity requires that:
a.
the objective function cannot be less than zero
Incorrect. Refer to Study Unit 1, Chapter 2.
b.
the decision variables cannot be less than zero
Correct.
c.
the right-hand side of the constraints cannot be greater than zero
Incorrect. Refer to Study Unit 1, Chapter 2.
d.
the reduced cost cannot be less than zero
Incorrect. Refer to Study Unit 1, Chapter 2.
7.
If a manufacturing process takes 4 hours per unit of x and 2 hours per unit of y and a
maximum of 100 hours of manufacturing process time is available, then an algebraic
formulation of this constraint is:
a.
4x + 2y ≥ 100
Incorrect. Refer to Study Unit 1, Chapter 2.
b.
4x − 2y ≤ 100
Incorrect. Refer to Study Unit 1, Chapter 2.
c.
4x + 2y ≤ 100
Correct.
d.
4x − 2y ≥ 100
Incorrect. Refer to Study Unit 1, Chapter 2.
8.
The feasible region in all linear programming problems is bounded by:
a.
corner points
Incorrect. Refer to Study Unit 1, Chapter 2.
SU1-36
BUS107
Introduction to Quantitative Analysis and Linear Programming
b.
hyperplanes
Correct.
c.
an objective line
Incorrect. Refer to Study Unit 1, Chapter 2.
d.
all of these options
Incorrect. Refer to Study Unit 1, Chapter 2.
9.
Suppose a company sells two different products, x and y, for net profits of $6 per unit
and $3 per unit, respectively. The slope of the line representing the objective function
is:
a.
0.5
Incorrect. Refer to Study Unit 1, Chapter 2.
b.
−0.5
Incorrect. Refer to Study Unit 1, Chapter 2.
c.
2
Incorrect. Refer to Study Unit 1, Chapter 2.
d.
−2
Correct.
10. The equation of the line representing the constraint 4x + 2y ≤ 100 passes through the
points:
a.
(25,0) and (0,50)
Correct.
b.
(0,25) and (50,0)
Incorrect. Refer to Study Unit 1, Chapter 2.
c.
(−25,0) and (0,−50)
SU1-37
BUS107
Introduction to Quantitative Analysis and Linear Programming
Incorrect. Refer to Study Unit 1, Chapter 2.
d.
(0,−25) and (−50,0)
Incorrect. Refer to Study Unit 1, Chapter 2.
11. When the profit increases with a unit increase in a resource, this change in profit will
be shown in the Solver's sensitivity report as the:
a.
add-in price
Incorrect. Refer to Study Unit 1, Chapter 2.
b.
sensitivity price
Incorrect. Refer to Study Unit 1, Chapter 2.
c.
shadow price
Correct.
d.
additional profit
Incorrect. Refer to Study Unit 1, Chapter 2.
12. Linear programming models have three important properties. They are:
a.
optimality, additivity and sensitivity
Incorrect. Refer to Study Unit 1, Chapter 2.
b.
optimality, linearity and divisibility
Incorrect. Refer to Study Unit 1, Chapter 2.
c.
divisibility, linearity and nonnegativity
Incorrect. Refer to Study Unit 1, Chapter 2.
d.
proportionality, additivity and divisibility
Correct.
13. Consider the following linear programming problem:
Maximize 4
+2
SU1-38
BUS107
Introduction to Quantitative Analysis and Linear Programming
Subject to:
4
+2
2
+
,
≤ 40
≥ 20
≥0
The above linear programming problem:
a.
has only one feasible solution
Incorrect. Refer to Study Unit 1, Chapter 2.
b.
has more than one optimal solution
Incorrect. Refer to Study Unit 1, Chapter 2.
c.
exhibits infeasibility
Correct.
d.
exhibits unboundedness
Incorrect. Refer to Study Unit 1, Chapter 2.
14. In a linear programming model, if the constraint’s Right-Hand-Side value exceeds
the range of feasibility, then:
a.
the dual price will remain the same.
Incorrect. Refer to Study Unit 1, Chapter 2.
b.
the dual price might change.
Incorrect. Refer to Study Unit 1, Chapter 2.
c.
the constraint will be unbounded.
Incorrect. Refer to Study Unit 1, Chapter 2.
d.
there will be no feasible solution.
Correct.
SU1-39
BUS107
Introduction to Quantitative Analysis and Linear Programming
References
Winston, W. L., & Albright, S. C. (2019). Practical management science (Sixth ed.). Cengage
Learning.
SU1-40
Study
Unit
2
Forecasting and Decision Analysis
BUS107
Forecasting and Decision Analysis
Learning Outcomes
By the end of this unit, you should be able to:
1.
Understand that the long-run success of an organisation is often closely related
to how well management is able to predict future aspects of the operation.
2.
Know the various components of a time series.
3.
Use smoothing techniques such as moving averages and exponential smoothing.
4.
Use the least square method to identify the trend component of a time series.
5.
Understand how the classical time series model can be used to explain the pattern
or behaviour of the data in a time series and to develop a forecast for the time
series.
6.
Determine and use seasonal indexes for a time series.
7.
Define a problem situation in terms of decisions to be made, chance events and
consequences.
8.
Explain what a decision strategy is.
9.
Identify a simple decision analysis problem from both a payoff table and a
decision tree point of view.
10.
Use a risk analysis and a sensitivity analysis to study how changes in problem
inputs affect or alter the recommended decision.
11.
Describe a Bayesian approach to computing revised branch probabilities.
12.
Identify the best decision under uncertainty (without probabilities) decision
making approaches: optimistic, conservative and minimax regret.
13.
Determine the best decision under risk (with probabilities) decision making
approaches: maximisation of Expected Value (EV) and Expected Value of Perfect
Information (EVPI).
14.
Illustrate how new information and revised probability values can be used in the
decision analysis approach to problem solving.
SU2-2
BUS107
Forecasting and Decision Analysis
Overview
Many decision-making applications depend on a forecast of some quantity. For a timebased forecast, where the interest is in future events, such forecast is often obtained
through an extrapolation forecasting method. For non-time based forecast where the
interest is in the outcome of an event, a decision analysis is the technique that provides
a framework and methodology for rational decision making when the outcomes are
uncertain.
Chapter 1 describes the fundamentals of forecasting. It introduces the techniques such as
moving average, exponential smoothing, a trend and seasonal analysis of historical data
concerning one or more time series.
Chapter 2 provides an introduction to decision analysis. It also examines decision making
techniques with and without probabilities.
SU2-3
BUS107
Forecasting and Decision Analysis
Chapter 1: Forecasting
Many decision-making applications depend on a forecast of some quantity. Forecasting
is simply a prediction of what will happen in the future. We must realise that, regardless
of the technique used, we will not be able to develop perfect forecasts. This chapter
introduces time-series forecasting techniques.
Read
You should now read Winston and Albright (2019), pp.715-718.
1.1 Time Series & Smoothing Methods in Forecasting
1.1.1 Time Series
A time series is a set of observations measured at successive points in time or over
successive periods of time.
If the historical data used are restricted to past values of the series that we are trying to
forecast, the procedure is called a time series method.
If the historical data used involve other time series that are believed to be related to the
time series that we are trying to forecast, the procedure is called a causal method.
Three time series methods are:
• smoothing
• trend projection
• trend projection adjusted for seasonal influence
SU2-4
BUS107
Forecasting and Decision Analysis
1.1.2 Components of Time Series
Trend - The trend component accounts for the gradual shifting of the time series over a
long period of time.
Cyclical - Any regular pattern of sequences of values above and below the trend line is
attributable to the cyclical component of the series.
Seasons - The seasonal component of the series accounts for regular patterns of variability
within certain time periods, such as over a year.
Irregularities (noise) - The irregular component of the series is caused by short-term,
unanticipated and non-recurring factors that affect the values of the time series. One
cannot attempt to predict its impact on the time series in advance.
Lesson Recording
Time Series & Smoothing Methods in Forecasting – Components of Time Series
1.1.3 Forecast Accuracy
Mean Squared Error: The average of the squared forecast errors for the historical data is
calculated. The forecasting method or parameter(s) which minimises this mean squared
error is then selected.
Mean Absolute Deviation: The mean of the absolute values of all forecast errors is
calculated, and the forecasting method or parameter(s) which minimises this measure is
selected. The mean absolute deviation measure is less sensitive to individual large forecast
errors than the mean squared error measure.
SU2-5
BUS107
Forecasting and Decision Analysis
Read
You should now read Winston and Albright (2019), pp.745-746.
1.1.4 Smoothing Methods
In cases in which the time series is fairly stable and has no significant trend, seasonal
components, or cyclical effects, one can use smoothing methods to average out the
irregular components of the time series.
The two common smoothing methods are:
• Moving averages
• Exponential smoothing
1.1.5 Moving Averages
Moving Average Method
The moving average method consists of computing an average of the most recent n data
values for the series and using this average for forecasting the value of the time series for
the next period.
Read
You should now read Winston and Albright (2019), pp.746-751.
SU2-6
BUS107
Forecasting and Decision Analysis
Lesson Recording
Time Series & Smoothing Methods in Forecasting – Moving Average & Centered
Moving Average
Centred Moving Average Method
The centred moving average method consists of computing an average of n periods' data
and associating it with the midpoint of the periods. For example, the average for periods 5,
6, and 7 is associated with period 6. This methodology is useful in the process of computing
season indexes.
Weighted Moving Average Method
In the weighted moving average method for computing the average of the most recent
n-periods, the more recent observations are typically given more weight than older
observations. For convenience, the weights usually sum to 1.
1.1.6 Exponential Smoothing
Using exponential smoothing, the forecast for the next period is equal to the forecast for
the current period plus a proportion (α) of the forecast error in the current period.
Using exponential smoothing, the forecast is calculated by:
α*[the actual value for the current period] + (1- α) *[the forecasted value for the current
period]
where the smoothing constant, α, is a number between 0 and 1.
SU2-7
BUS107
Forecasting and Decision Analysis
Read
You should now read Winston and Albright (2019), pp.751-755.
Lesson Recording
Time Series & Smoothing Methods in Forecasting – Exponential Smoothing
1.1.7 Worked Example
Moving averages often are used to identify movements in stock prices. Daily closing prices
(in dollars per share) for SIM Pte Ltd for January 5, 2015, through January 20, 2015, are as
follows:
Table 2.1
Day
Price ($)
Day
Price ($)
January 5
14.45
January 13
16.45
January 6
15.75
January 14
15.60
January 7
16.45
January 15
15.09
January 8
17.40
January 16
16.42
January 9
17.32
January19
16.21
January 12
15.96
January20
15.22
SU2-8
BUS107
Forecasting and Decision Analysis
a.
Use a five-day moving average to smooth the time series. Forecast the closing
price for January 21, 2015 and the Mean Squared Error (MSE) for this technique.
Since it is stated to use a five-day period for the computation of moving average,
this means that the input to the first forecast is made up of the first five data from
January 5 to January 9. This computed forecast will be for the next period, that
is, for January 12.
Forecast (January 12, Day 6) = (14.45 + 15.75 + 16.45 + 17.40 + 17.32) / 5 = 16.27
Since the actual value for January 12 is available ($15.96), we can compute the
forecast error:
Forecast Error (January 12, Day 6) = 15.96 – 16.27 = -0.31
We repeat the steps above to obtain the following consolidated result as shown
in Table 2.2.
Table 2.2
Day
Time-
5-Day
Forecast
Series Value
Moving
Error
Average
Forecast
1
14.45
2
15.75
3
16.45
4
17.40
5
17.32
6
15.96
16.27
– 0.31
0.10
7
16.45
16.58
– 0.13
0.02
SU2-9
BUS107
Forecasting and Decision Analysis
Day
Time-
5-Day
Forecast
Series Value
Moving
Error
Average
Forecast
8
15.60
16.72
– 1.12
1.25
9
15.09
16.55
– 1.46
2.13
10
16.42
16.08
0.34
0.12
11
16.21
15.90
0.31
0.10
12
15.22
15.95
– 0.73
0.53
Total:
4.25
Hence,
MSE = 4.25/7 = 0.61
Forecast (January 20, Day 13) = (15.60 + 15.09 + 16.42 + 16.21 + 15.22)/5 = 15.71
b.
Use exponential smoothing with a smoothing constant of α = 0.7 to smooth the
time series. Forecast the closing price for January 21, 2015 and the MSE for this
technique.
Let:
= Actual value in Day
= Forecasted sales in Day
The first forecast value in exponential smoothing is either provided or to simply
use the previous period actual:
Forecast (January 6, Day 2) =
Then,
=α
+ (1-α)
SU2-10
=
= 14.45
BUS107
Forecasting and Decision Analysis
= 0.7
i.e.
+ 0.3
Forecast (January 7, Day 3) = 0.7x15.75 + 0.3x14.45 = 15.36
Table 2.3
Day
Time-
Forecast
Series Value
Forecast
Error
1
14.45
2
15.75
14.45
1.3
1.69
3
16.45
15.36
1.09
1.19
4
17.40
16.12
1.28
1.64
5
17.32
17.02
0.3
0.09
6
15.96
17.23
– 1.27
1.61
7
16.45
16.34
0.11
0.01
8
15.60
16.42
– 0.82
0.67
9
15.09
15.85
– 0.76
0.58
10
16.42
15.32
1.1
1.21
11
16.21
16.09
0.12
0.01
12
15.22
16.17
– 0.95
0.90
Total:
9.57
Hence,
MSE = 9.57/11 = 0.87
Forecast (January 20, Day 13) = 0.7(15.22) + 0.3(16.17) = 15.51
Moving Averages approach is the better of the two approaches because it has the
smallest MSE.
SU2-11
BUS107
Forecasting and Decision Analysis
1.2 Trend Projection and Seasonal Components
1.2.1 Trend Projection
If a time series exhibits a linear trend, the method of least squares may be used to
determine a trend line (projection) for future forecasts.
Least squares, also used in a regression analysis, determines the unique trend line forecast
which minimises the mean square error between the trend line forecasts and the actual
observed values for the time series.
The independent variable is the time period and the dependent variable is the actual
observed value in the time series.
Using the method of least squares, the formula for the trend projection is:
=
+
t
where:
= trend forecast for time period ,
= slope of the trend line, and
= trend line projection for time 0.
where:
= observed value of the time series at time period ,
= average of the observed values for
= average time period for the
, and
observations.
SU2-12
BUS107
Forecasting and Decision Analysis
Read
You should now read Winston and Albright (2019), pp.717-724.
Lesson Recording
Time Series & Smoothing Methods in Forecasting – Trend Projection
1.2.2 Forecasting with Trend and Seasonal Components
When a time series exhibits obvious seasonality, such as swimming pool supply sales
that are always higher in the spring and summer than in the rest of the year, none of the
extrapolation methods discussed to this point does a good job. They all miss the seasonal
ups and downs.
There are several methods that deal with seasonality. This study guide presents the
multiplicative time series model while the course textbook presents the Winters’ method
for seasonality.
Multiplicative Time Series Model
These are the steps of multiplicative time series model:
1.
Calculate the centred moving averages (CMAs).
2.
Centre the CMAs on integer-valued periods.
3.
Determine the seasonal and irregular factors (
4.
Determine the average seasonal factors.
5.
Scale the seasonal factors (
6.
Determine the deseasonalised data.
).
SU2-13
).
BUS107
Forecasting and Decision Analysis
7.
Determine a trend line of the deseasonalised data.
8.
Determine the deseasonalised predictions.
9.
Take into account the seasonality.
Lesson Recording
Time Series & Smoothing Methods in Forecasting – Seasonal Components
Winters’ Method for Seasonality
Winters’ method is a direct extension of Holt’s exponential smoothing model.
Read
You should now read Winston and Albright (2019), pp.758-760.
Lesson Recording
Time Series & Smoothing Methods in Forecasting – Trend and Seasonal Components
SU2-14
BUS107
Forecasting and Decision Analysis
Activity 1
Discuss the purpose of the deseasonalisation step in a multiplicative time series
model.
Review Questions
1.
What are the three groups that forecasting models can be divided into?
2.
Identify the condition needed when interpreting the coefficient for a particular
independent variable X in a multiple regression equation.
3.
What are commonly used summary measures for forecast errors?
4.
Identify the different components in a time series forecasting model.
SU2-15
BUS107
Forecasting and Decision Analysis
Chapter 2: Decision Analysis
This chapter discusses methods that can be used in decision-making problems where
uncertainty is a key element. This chapter approaches decision-making problems with
uncertainty in a systematic manner.
Read
You should now read Winston and Albright (2019), pp.457-463.
2.1 Problem Formulation and Decision-Making without
Probabilities
Although decision-making under uncertainty occurs in a wide variety of contexts, the
problems are alike in the following ways:
• A problem has been identified that requires a solution.
• A number of possible decisions have been identified.
• Each decision leads to a number of possible outcomes.
• There is uncertainty about which outcome will occur.
One of the main reasons why decision-making under uncertainty is difficult is that
decisions have to be made before uncertain outcomes are revealed. For example, you must
place your bet at a roulette wheel before the wheel is spun. Before you make a decision,
you must at least list the possible outcomes that might occur.
2.1.1 Payoffs
Decisions and outcomes have consequences, either good or bad. These must be assessed
before intelligent decisions can be made. In our problems, these will be monetary payoffs
SU2-16
BUS107
Forecasting and Decision Analysis
or costs, but in many real-world decision problems, they can be nonmonetary, such as
environmental damage or loss of life.
Once all of these elements of a decision problem have been specified, it is time to make
some difficult trade-offs.
For example, would you rather take a chance at receiving $1 million, with the risk of losing
$2 million, or would you rather play it safer?
If very large amounts of money are at stake (relative to your wealth), your attitude towards
risk can also play a key role in the decision-making process.
Lesson Recording
Problem Formulation and Decision Making Without Probabilities – Payoff Table
2.1.2 Three Approaches
Three commonly used criteria for decision-making when probability information
regarding the likelihood of the states of nature is unavailable are:
• Optimistic approach (or Maximax)
• Conservative approach (or Maximin)
• Minimax Regret approach.
Optimistic Approach:
This approach is also known as “Choose the best of the best” approach, and has the
following traits:
• The optimistic approach would be used by an optimistic decision maker.
• The decision with the largest possible payoff is chosen.
SU2-17
BUS107
Forecasting and Decision Analysis
• If the payoff table was in terms of costs, the decision with the lowest cost would
be chosen.
Lesson Recording
Problem Formulation and Decision Making Without Probabilities – Optimistic
Approach
Conservative Approach:
This approach is also known as “Choose the best among the worst” approach, and has
the following traits:
• The conservative approach would be used by a conservative decision maker.
• For each decision the minimum payoff is listed and then the decision corresponding
to the maximum of these minimum payoffs is selected.
• If the payoff was in terms of costs, the maximum costs would be determined for each
decision and then the decision corresponding to the minimum of these maximum
costs is selected.
Lesson Recording
Problem Formulation and Decision Making Without Probabilities – Conservative
Approach
Minimax Regret Approach:
This approach is a compromise of the Optimistic and Conservative approach, and has the
following traits:
SU2-18
BUS107
Forecasting and Decision Analysis
• The minimax regret approach requires the construction of a regret table or an
opportunity loss table.
• This is done by calculating for each state of nature the difference between each
payoff and the largest payoff for that state of nature.
• Then, using this regret table, the maximum regret for each possible decision is listed.
• The decision chosen is the one corresponding to the minimum of the maximum
regrets.
Lesson Recording
Problem Formulation and Decision Making Without Probabilities – MiniMax Regret
Approach
2.2 Decision-Making with Probabilities
When probabilities information are made available for each uncertainty, then the
appropriate tool to use for analysis is the decision tree. A decision tree enables a decision
maker to view all important aspects of the problem at once: the decision alternatives,
the uncertain outcomes and their probabilities, the economic consequences, and the
chronological order of events.
As a decision maker, you must also assess the likelihoods of these outcomes with
probabilities. Note that these outcomes are generally not equally likely.
There is no easy way to assess the probabilities of the possible outcomes. Sometimes they
will be determined at least partly by historical data.
SU2-19
BUS107
Forecasting and Decision Analysis
2.2.1 Decision Trees
You need a decision criterion for choosing between two or more probability distributions
of payoff/cost outcomes. The decision problem we have been analysing is very basic.
• You make a decision, you then observe an outcome, you receive a payoff, and that
is the end of it.
• Many decision problems are of this basic form, but many are more complex.
• In these more complex problems, you make a decision, you observe an outcome,
you make a second decision, you observe a second outcome, and so on.
A graphical tool called a decision tree has been developed to represent decision problems.
Since in most situations the problem involves monetary outcomes, the decision criterion
used in decision tree is the expected monetary value, or EMV, criterion.
Some facts on decision trees:
1.
Decision trees are composed of nodes (circles, squares, and triangles) and
branches (lines).
2.
The nodes represent points in time. A decision node (a square) represents a
time when the decision maker makes a decision. A probability node (a circle)
represents a time when the result of an uncertain outcome becomes known. An
end node (a triangle) indicates that the problem is completed − all decisions have
been made, all uncertainty has been resolved, and all payoffs and costs have been
incurred.
3.
Time proceeds from left to right. This means that any branches leading into a
node (from the left) have already occurred. Any branches leading out of a node
(to the right) have not yet occurred.
4.
Branches leading out of a decision node represent the possible decisions;
the decision maker can choose the preferred branch. Branches leading out
of probability nodes represent the possible outcomes of uncertain events; the
decision maker has no control over which of these will occur.
SU2-20
BUS107
Forecasting and Decision Analysis
5.
Probabilities are listed on probability branches. These probabilities are
conditional on the events that have already been observed (those to the left).
Also, the probabilities on branches leading out of any probability node must sum
to 1.
6.
Monetary values (or payoff values) are shown to the right of the end nodes.
7.
EMVs are calculated through a “folding-back” process, discussed next. They
are shown above the various nodes. It is then customary to mark the optimal
decision branch(es) in some way. We have marked ours with a small notch.
Decision trees allow you to use a folding-back procedure to find the EMVs and the
optimal decision.
Read
You should now read Winston and Albright (2019), pp.465-471.
Lesson Recording
Decision Making With Probabilities – Decision Tree
2.2.2 Expected Value of Perfect Information
The expected value of perfect information (EVPI), or value of information, is the increase
in the expected profit that would result if one knew with certainty which state of nature
would occur.
EVPI = EVwPI – Max EMV
where
SU2-21
BUS107
Forecasting and Decision Analysis
EVwPI = (Best outcome for the 1st state of nature)*(Probability of the 1st state of nature)
+ (Best outcome for the 2nd state of nature)*(Probability of the 2nd state of nature) +…+
(Best outcome for the last state of nature)*(Probability of the last state of nature).
Lesson Recording
Decision Making With Probabilities – Expected Value Approach and Expected Value
of Perfect Information
Read
You should now read Winston and Albright (2019), pp.486-488.
2.2.3 Risk Analysis and Risk Profile
Risk analysis helps the decision maker recognise the difference between:
• the expected value of a decision alternative, and
• the payoff that might actually occur.
The risk profile for a decision alternative shows the possible payoffs for the decision
alternative along with their associated probabilities.
It is simply a plot of Payoffs v Probabilities for each decision alternative.
SU2-22
BUS107
Forecasting and Decision Analysis
Lesson Recording
Decision Making With Probabilities – Risk Profile
Read
You should now read Winston and Albright (2019), pp.476-478.
Activity 2
Rational decision makers are sometimes willing to violate the EMV maximisation
criterion when large amounts of money are at stake. Discuss the validity of this
statement.
Review Questions
1.
What are the three common elements related to decision-making under uncertainty?
2.
Define the expected value of information.
3.
A customer has approached a local credit union for a $20,000 1-year loan at a 10%
interest rate. If the credit union does not approve the loan application, the $20,000 will
be invested in bonds that earn a 6% annual return. Without additional information,
the credit union believes that there is a 5% chance that this customer will default on
the loan, assuming that the loan is approved. If the customer defaults on the loan, the
credit union will lose the $20,000. Construct a decision tree to help the credit union
SU2-23
BUS107
Forecasting and Decision Analysis
decide whether or not to make the loan. Make sure to label all decision and chance
nodes and include appropriate costs, payoffs and probabilities.
SU2-24
BUS107
Forecasting and Decision Analysis
Summary
In real situations, the data is often obtained through a regression or an extrapolation
forecasting method. In Chapter 1 of this Study Unit, we have discussed regression analyses
and some of the more popular extrapolation methods for time series forecasting.
Chapter 2 of this Study Unit provides a formal framework for analysing decision problems
that involve uncertainty with discussion that includes the following: the criteria for
choosing among alternative decisions, how probabilities are used in the decision-making
process, how early decisions affect decisions made at a later stage, how a decision maker
can quantify the value of information, how attitudes towards risk can affect the analysis.
SU2-25
BUS107
Forecasting and Decision Analysis
Formative Assessment
1.
All problems related to decision making under uncertainty have three common
elements:
a. the mean, median, and mode
b. the set of decisions, the cost of each decision and the profit that can be made
from each decision
c. the set of possible outcomes, the set of decision variables and the constraints
d. the set of decisions, the set of possible outcomes, and a value model that
prescribes results
2.
Expected monetary value (EMV) is:
a. the average or expected value of the decision if you knew what would happen
ahead of time
b. the weighted average of possible monetary values, weighted by their
probabilities
c. the average or expected value of the information if it was completely accurate
d. the amount that you would lose by not picking the best alternative
3.
Probabilities on the branches of a chance node may be ____ events that have occurred
earlier in the decision tree.
a. marginal due to
b. conditional on
c. averaged with
d. increased by
4.
Which of the following statements is true concerning decision tree conventions?
a. Time proceeds from right to left.
b. The trees are composed of circles, triangles and ovals.
SU2-26
BUS107
Forecasting and Decision Analysis
c. The nodes represent points in time.
d. Probabilities of outcomes are shown to the right of the end nodes.
5.
The solution procedure for solving decision trees is called:
a. sensitivity analysis
b. policy iteration
c. risk profiling
d. folding back
6.
The strategy region graph is a type of sensitivity analysis chart that:
a. is useful in determining whether the optimal decision changes over the range
of the input variable.
b. ranks the sensitivity of the EMV to the input variables.
c. reflects how the value of information changes over a range of probabilities.
d. None of these
7.
Bayes’ rule is used to:
a. update the prior probabilities once new information is observed.
b. turn the given conditional probabilities (i.e. likelihoods) around.
c. update the posterior probabilities once new information is observed.
d. All of the above are uses for Bayes’ rule.
8.
The denominator of Bayes' rule:
a. is the same as the simple probability of an outcome O.
b. decomposes the probability of the new information I into all possibilities.
c. is sometimes called the law of complementary probabilities.
d. is unique for each possible outcome.
9.
Which of the following are probabilities that are conditioned on information that is
obtained?
SU2-27
BUS107
Forecasting and Decision Analysis
a. Prior probabilities
b. Posterior probabilities
c. Marginal probabilities
d. Objective probabilities
10. A utility function for risk averse individuals is ____ and/or ____.
a. decreasing, linear
b. decreasing, convex
c. increasing, linear
d. increasing, concave
11. Which of the following is not one of the commonly used summary measures for
forecast errors?
a. MAE (mean absolute error)
b. MFE (mean forecast error)
c. MSE (mean square error)
d. MAD (mean absolute deviation)
12. When using the moving average method, you must select ____ which represent(s) the
number of terms in the moving average.
a. a smoothing constant
b. the explanatory variables
c. an alpha value
d. a span
13. In exponential smoothing, ____ represents the weightage placed on the actual value
for the current period.
a. a smoothing constant
b. the explanatory variables
c. the standard deviation
SU2-28
BUS107
Forecasting and Decision Analysis
d. the sample mean
SU2-29
BUS107
Forecasting and Decision Analysis
Solutions or Suggested Answers
Chapter 1 Review Questions
1.
What are the three groups that forecasting models can be divided into?
Judgemental, regression, and extrapolation methods.
2.
Identify the condition needed when interpreting the coefficient for a particular
independent variable X in a multiple regression equation.
An important condition when interpreting the coefficient for a particular
independent variable X in a multiple regression equation is that all of the other
independent variables remain constant.
3.
What are commonly used summary measures for forecast errors?
These are the commonly used summary measures for forecast errors.
• MAE (mean absolute error)
• MSE (mean square error)
• MAPE (mean absolute percentage error)
4.
Identify the different components in a time series forecasting model.
These are the four components in a time series forecasting model:
• Trend,
• Seasonal,
• Cyclical, and
• Irregularities (noise).
SU2-30
BUS107
Forecasting and Decision Analysis
Chapter 2 Review Questions
1.
What are the three common elements related to decision-making under uncertainty?
The set of decisions, the set of possible outcomes, and a value model that prescribes
results.
2.
Define the expected value of information.
The expected value of information (EVI) is the difference between the EMV
obtained with free sample information and the EMV obtained without any
information.
3.
A customer has approached a local credit union for a $20,000 1-year loan at a 10%
interest rate. If the credit union does not approve the loan application, the $20,000 will
be invested in bonds that earn a 6% annual return. Without additional information,
the credit union believes that there is a 5% chance that this customer will default on
the loan, assuming that the loan is approved. If the customer defaults on the loan, the
credit union will lose the $20,000. Construct a decision tree to help the credit union
decide whether or not to make the loan. Make sure to label all decision and chance
nodes and include appropriate costs, payoffs and probabilities.
Formative Assessment
1.
All problems related to decision making under uncertainty have three common
elements:
a.
the mean, median, and mode
SU2-31
BUS107
Forecasting and Decision Analysis
Incorrect. Refer to Study Unit 2, Chapter 2.
b.
the set of decisions, the cost of each decision and the profit that can be made
from each decision
Incorrect. Refer to Study Unit 2, Chapter 2.
c.
the set of possible outcomes, the set of decision variables and the constraints
Incorrect. Refer to Study Unit 2, Chapter 2.
d.
the set of decisions, the set of possible outcomes, and a value model that
prescribes results
Correct.
2.
Expected monetary value (EMV) is:
a.
the average or expected value of the decision if you knew what would
happen ahead of time
Incorrect. Refer to Study Unit 2, Chapter 2.
b.
the weighted average of possible monetary values, weighted by their
probabilities
Correct.
c.
the average or expected value of the information if it was completely accurate
Incorrect. Refer to Study Unit 2, Chapter 2.
d.
the amount that you would lose by not picking the best alternative
Incorrect. Refer to Study Unit 2, Chapter 2.
3.
Probabilities on the branches of a chance node may be ____ events that have occurred
earlier in the decision tree.
a.
marginal due to
Incorrect. Refer to Study Unit 2, Chapter 2.
b.
conditional on
SU2-32
BUS107
Forecasting and Decision Analysis
Correct!
c.
averaged with
Incorrect. Refer to Study Unit 2, Chapter 2.
d.
increased by
Incorrect. Refer to Study Unit 2, Chapter 2.
4.
Which of the following statements is true concerning decision tree conventions?
a.
Time proceeds from right to left.
Incorrect. Refer to Study Unit 2, Chapter 2.
b.
The trees are composed of circles, triangles and ovals.
Incorrect. Refer to Study Unit 2, Chapter 2.
c.
The nodes represent points in time.
Correct!
d.
Probabilities of outcomes are shown to the right of the end nodes.
Incorrect. Refer to Study Unit 2, Chapter 2.
5.
The solution procedure for solving decision trees is called:
a.
sensitivity analysis
Incorrect. Refer to Study Unit 2, Chapter 2.
b.
policy iteration
Incorrect. Refer to Study Unit 2, Chapter 2.
c.
risk profiling
Incorrect. Refer to Study Unit 2, Chapter 2.
d.
folding back
Correct!
SU2-33
BUS107
6.
Forecasting and Decision Analysis
The strategy region graph is a type of sensitivity analysis chart that:
a.
is useful in determining whether the optimal decision changes over the range
of the input variable.
Correct!
b.
ranks the sensitivity of the EMV to the input variables.
Incorrect. Refer to Study Unit 2, Chapter 2.
c.
reflects how the value of information changes over a range of probabilities.
Incorrect. Refer to Study Unit 2, Chapter 2.
d.
None of these
Incorrect. Refer to Study Unit 2, Chapter 2.
7.
Bayes’ rule is used to:
a.
update the prior probabilities once new information is observed.
Incorrect. Refer to Study Unit 2, Chapter 2.
b.
turn the given conditional probabilities (i.e. likelihoods) around.
Incorrect. Refer to Study Unit 2, Chapter 2.
c.
update the posterior probabilities once new information is observed.
Incorrect. Refer to Study Unit 2, Chapter 2.
d.
All of the above are uses for Bayes’ rule.
Correct!
8.
The denominator of Bayes' rule:
a.
is the same as the simple probability of an outcome O.
Incorrect. Refer to Study Unit 2, Chapter 2.
b.
decomposes the probability of the new information I into all possibilities.
Correct!
SU2-34
BUS107
Forecasting and Decision Analysis
c.
is sometimes called the law of complementary probabilities.
Incorrect. Refer to Study Unit 2, Chapter 2.
d.
is unique for each possible outcome.
Incorrect. Refer to Study Unit 2, Chapter 2.
9.
Which of the following are probabilities that are conditioned on information that is
obtained?
a.
Prior probabilities
Incorrect. Refer to Study Unit 2, Chapter 2.
b.
Posterior probabilities
Correct!
c.
Marginal probabilities
Incorrect. Refer to Study Unit 2, Chapter 2.
d.
Objective probabilities
Incorrect. Refer to Study Unit 2, Chapter 2.
10. A utility function for risk averse individuals is ____ and/or ____.
a.
decreasing, linear
Incorrect. Refer to Study Unit 2, Chapter 2.
b.
decreasing, convex
Incorrect. Refer to Study Unit 2, Chapter 2.
c.
increasing, linear
Incorrect. Refer to Study Unit 2, Chapter 2.
d.
increasing, concave
Correct!
SU2-35
BUS107
Forecasting and Decision Analysis
11. Which of the following is not one of the commonly used summary measures for
forecast errors?
a.
MAE (mean absolute error)
Incorrect. Refer to Study Unit 2, Chapter 1.
b.
MFE (mean forecast error)
Correct!
c.
MSE (mean square error)
Incorrect. Refer to Study Unit 2, Chapter 1.
d.
MAD (mean absolute deviation)
Incorrect. Refer to Study Unit 2, Chapter 1.
12. When using the moving average method, you must select ____ which represent(s) the
number of terms in the moving average.
a.
a smoothing constant
Incorrect. Refer to Study Unit 2, Chapter 1.
b.
the explanatory variables
Incorrect. Refer to Study Unit 2, Chapter 1.
c.
an alpha value
Incorrect. Refer to Study Unit 2, Chapter 1.
d.
a span
Correct!
13. In exponential smoothing, ____ represents the weightage placed on the actual value
for the current period.
a.
a smoothing constant
Correct!
b.
the explanatory variables
SU2-36
BUS107
Forecasting and Decision Analysis
Incorrect. Refer to Study Unit 2, Chapter 1.
c.
the standard deviation
Incorrect. Refer to Study Unit 2, Chapter 1.
d.
the sample mean
Incorrect. Refer to Study Unit 2, Chapter 1.
SU2-37
BUS107
Forecasting and Decision Analysis
References
Winston, W. L., & Albright, S. C. (2019). Practical management science (Sixth ed.). Cengage
Learning.
SU2-38
Study
Unit
3
Simulation and Network Modelling
BUS107
Simulation and Network Modelling
Learning Outcomes
By the end of this unit, you should be able to:
1.
define what simulation is, and its applications to a variety of situations in the
analysis of problems.
2.
list the advantages and disadvantages of simulation.
3.
discuss the use of simulation to perform a risk analysis to predict the outcome
of a decision under uncertainty.
4.
identify the important role that probability distributions, random numbers, and
the computer play in implementing simulation models.
5.
describe the three network techniques: the minimal spanning tree model, the
maximal flow model and the shortest-route model.
6.
develop the network and solve the shortest-route problems stating procedures
to obtain the optimal solution.
7.
develop the network and solve the maximal flow problems stating iterations to
obtain the optimal solution.
8.
solve the minimal spanning tree problems by connecting all nodes using arcs.
SU3-2
BUS107
Simulation and Network Modelling
Overview
Simulation is one of the most widely used quantitative methods in decision-making.
It is about learning a real system by experimenting (or simulating) with a model that
represents the system. For physical systems, many important optimisation models have a
natural graphical network representation.
Chapter 1 illustrates the basic ideas of simulation. Simulation models provide important
insights that are missing in models that do not incorporate uncertainty explicitly.
Chapter 2 discusses some specific examples of network models. In particular, the shortestroute (also known as shortest-path) model, maximal flow model and minimal spanning
tree model will be covered.
SU3-3
BUS107
Simulation and Network Modelling
Chapter 1: Simulation Modelling
This chapter discusses methods that can be used in decision-making problems where
uncertainty is a key element. This chapter approaches decision problems with uncertainty
in a systematic manner.
Read
You should now read Winston and Albright (2019), pp.515-518.
1.1 Simulation Modelling and Applications
The world is full of uncertainty, which is what makes simulation so relevant and useful.
It is about learning a real system by experimenting (or simulating) with a model that
represents the system.
Possible Applications:
Financial Forecasting – predicting the future values of a product based on its previous
performance (e.g. stocks).
New Product Development – predicting the profitability of a product based on its demand
and costs (parts & labour).
Inventory Control – simulation is widely used here to determine product demand for
inventory control.
Waiting Lines – simulation on customers’ waiting time and service time.
SU3-4
BUS107
Simulation and Network Modelling
Lesson Recording
Simulation Modelling and Applications – Purpose of Simulation
1.1.1 Probabilities and Simulation
A simulation model contains both input and output variables. Mathematical expressions
and logical relationships are used to define input and output variables. Probability and
random numbers are used to simulate the results.
Read
You should now read Winston and Albright (2019), pp.518-530.
1.1.2 Monte Carlo Simulation
The basis of the Monte Carlo simulation is experimentation on the probabilistic elements
through random sampling. It is used with probabilistic variables.
The four-step approach in performing the Monte Carlo Simulation is as follows:
1.
Set up probability distribution of past performance.
2.
Establish an interval of random numbers.
3.
Generate random numbers.
4.
Perform simulation by mapping random number with input values.
SU3-5
BUS107
Simulation and Network Modelling
Lesson Recording
Simulation Modelling and Applications – Monte Carlo Simulation
For step 2, it is also possible to use the probabilities derived to set up the mapping interval.
Lesson Recording
Simulation Modelling and Applications – Monte Carlo Simulation using Probabilities
Interval
1.1.3 Worked Example
The number of cars sold in a given month by Performance Cars during the last 6 months
has been recorded as follows:
Table 3.1
Month
Number of Cars Sold
1
5
2
8
3
2
4
10
5
2
6
5
SU3-6
BUS107
Simulation and Network Modelling
Perform the Monte Carlo Simulation method to simulate the demand for the next 6 months
using random numbers 62, 32, 71, 94, 04 and 97 as the probability demand.
Step 1: Set up probability distribution of past performance
Table 3.2
Month
Number of Cars Sold
Probability of Demand
1
5
0.16
2
8
0.25
3
2
0.06
4
10
0.31
5
2
0.06
6
5
0.16
Total
32
1.00
Step 2: Establish interval of random numbers
Since the random numbers provided are 2 digits, therefore the range of random numbers
to be assigned is 0 ~ 99, that is, a total of 100 unique numbers. (Hence, if you are provided
with a random number that comprises 3 digits, then the range of random numbers to be
assigned is 0 ~ 999, a total of 1000 unique numbers.)
For the first event (5 cars sold), its corresponding probability is 0.16. Hence, out of the 100
unique numbers, (0.16 * 100) = 16 numbers are allocated to this event; 0 ~ 15.
For the second event (8 cars sold), its corresponding probability is 0.25. Hence, out of the
100 unique numbers, (0.25 * 100) = 25 numbers are allocated to this event; 16 ~ 40. Thus,
SU3-7
BUS107
Simulation and Network Modelling
following the same logic, each event is allocated the probability apportioned number of
unique numbers.
Table 3.3
Month
Number of
Probability
Random
Cars Sold
of Demand
Interval Range
1
5
0.16
00 – 15
2
8
0.25
16 – 40
3
2
0.06
41 – 46
4
10
0.31
47 – 77
5
2
0.06
78 – 83
6
5
0.16
84 – 99
Total
32
1.00
Step 3: Generate random numbers
The random numbers provided are: 62, 32, 71, 94, 04 and 97
Step 4: Perform simulation by mapping random numbers with input values
Referring to Table 3.3, the random number 62 maps to the random interval 47~77, which
corresponds to the event of 10 cars being sold. Likewise, the random number 32 maps to
the random interval 16~40, which corresponds to the event of 8 cars being sold. Following
through the same mapping procedures for all the random numbers generated in Step 3,
we obtained the simulated result for the next 6 months as shown in Table 3.4.
SU3-8
BUS107
Simulation and Network Modelling
Table 3.4
Month
Random Number
Number of Cars Sold
7
62
10
8
32
8
9
71
10
10
94
5
11
04
5
12
97
5
Total
43
Read
You should now read Winston and Albright (2019), pp.540-545.
Activity 1
Discuss, from a general perspective, the advantages of simulation analysis.
Review Questions
1.
What is the Monte Carlo simulation? List the major steps in the Monte Carlo
simulation process.
2.
Simulation has been successfully applied in a variety of applications. List and discuss
two (2) of these areas of applications.
SU3-9
BUS107
3.
Simulation and Network Modelling
Discuss the advantages and disadvantages of using a simulation approach to a risk
analysis.
4.
Identify the reasons for using simulation as a technique.
SU3-10
BUS107
Simulation and Network Modelling
Chapter 2: Network Modelling
This chapter discusses methods that can be used in decision-making problems where
uncertainty is a key element. This chapter approaches decision problems with uncertainty
in a systematic manner.
Read
You should now read Winston and Albright (2019), pp.219-221.
2.1 Network Models
Many companies have real problems, often extremely large, that can be represented as
network models. In fact, many of the best management science success stories have
involved large network models. Specialised solution techniques have been developed
specifically for network models. The more commonly known network models include:
• Transportation model
• Assignment model
• Shortest-route (shortest-path) model
• Maximal (maximum) flow model
• Minimal (minimum) spanning tree
2.1.1 Transportation Model
In many situations, a company produces products at locations called origins and ships
these products to customer locations called destinations. Typically, each origin has a
limited amount that it can ship, and each customer destination must receive a required
quantity of the product. The assumption is that the only possible shipments are those
SU3-11
BUS107
Simulation and Network Modelling
directly from an origin to a destination. That is, no shipments between origins or between
destinations are possible.
A typical transportation problem requires three sets of data: capacities (or supplies),
demands (or requirements), and unit shipping (and possibly production) costs.
The capacities indicate the maximum amount that each plant can supply in a given
amount of time under current operating conditions. In some cases, it might be possible to
increase the “base” capacities, by using overtime, for example. In such cases, the model
could be modified to determine the amounts of additional capacity to use (and pay for).
The customer demands are typically estimated from some type of forecasting models. The
forecasts are often based on historical customer demand data.
The unit shipping costs come from a transportation cost analysis - what does it really cost
to send a single automobile from any plant to any region? The unit “shipping” costs can
also include the unit production cost at each plant.
Read
You should now read Winston and Albright (2019), pp.221-232.
2.1.2 Assignment Model
Assignment models are used to assign, on a one-to-one basis, members of one set to
members of another set in the least-cost (or the least-time) manner.
The prototype assignment model is the assignment of machines to jobs.
For example, suppose there are four jobs and five machines. Every pairing of a machine
and a job has a given job completion time. The problem is to assign the machines to the
jobs so that the total time to complete all jobs is minimised.
SU3-12
BUS107
Simulation and Network Modelling
Read
You should now read Winston and Albright (2019), pp.233-239.
2.1.3 Shortest-Route/Path Model
In many applications, the objective is to find the shortest path between two points in a
network.
Sometimes this problem occurs in a geographical context where, for example, the objective
is to find the shortest path on interstate freeways from Seattle to Miami. There are also
problems that do not look like the shortest-path problems but can be modelled in the same
way.
The typical shortest-path problem is a special case of the network flow problem from the
previous section.
To see why this is the case, suppose that you want to find the shortest path between node
1 and node N in a network.
To find this shortest path, you create a network flow model where the supply for node 1
is 1, and the demand for node N is 1. All other nodes are transshipment nodes.
Read
You should now read Winston and Albright (2019), pp.249-257.
There are several approaches to solve a shortest-route problem. This is one of those
methods:
SU3-13
BUS107
Simulation and Network Modelling
Step 1:
Assign node 1 the permanent label [0,S]. The first number is the distance from node 1; the
second number is the preceding node. Since node 1 has no preceding node, it is labelled
S for the starting node.
Step 2:
Assign tentative labels, (d,n), for the nodes that can be reached directly from node 1 where
d = the direct distance from node 1
n = the preceding node
Step 3:
Make a tentative label permanent and repeat steps 2 and 3.
Its applications include:
• Highway travel between cities
• New road construction
• Facility location
• Equipment replacement
Lesson Recording
Network Models – Shortest Route
2.1.4 Maximal Flow Model
The maximal flow problem is concerned with determining the maximal volume of flow
from one node (called the source) to another node (called the sink).
SU3-14
BUS107
Simulation and Network Modelling
For example, this technique can determine the maximum number of vehicles (cars, trucks,
etc.) that can go through a network of roads from one location to another.
Its applications include:
• Traffic flow systems
• Production line flows
• Shipping
Lesson Recording
Network Models – Maximum Flow
2.1.5 Minimal Spanning Tree Model
A spanning tree is a tree that connects all nodes of a network. The minimal spanning tree
problem seeks to determine the minimum sum of the arc lengths necessary to connect all
nodes in a network.
The criterion to be minimised in the minimal spanning tree problem is not limited to
distance even though the term "closest" is used in describing the procedure. Other criteria
include time and cost. (Neither time nor cost is necessarily linearly related to distance.)
In essence, it helps to find the minimum total distance that connects all nodes in the
network.
Its applications include:
• Sewer system design
• Computer system layout
• Cable television connections
• Mass transit design
SU3-15
BUS107
Simulation and Network Modelling
Lesson Recording
Network Models – Minimum Spanning Tree
Activity 2
Rational decision makers are sometimes willing to violate the EMV maximisation
criterion when large amounts of money are at stake. Discuss the validity of this
statement.
Review Questions
1.
Describe the characteristics of the “Shortest-path model” and list three (3) of its
potential applications.
2.
Describe the following network models and provide each with Two (2) examples of
their application.
• Transportation
• Assignment
• Traveling Salesman
3.
The fundamental underlying assumption in a transportation model methodology is
that the units of ‘product’ being transported are homogeneous. Discuss how you can
modify the model to achieve better fit.
4.
The figure below shows a network of connected expressways with the capacity given
on each section. What is the maximum possible flow between nodes 1 and 7?
SU3-16
BUS107
Simulation and Network Modelling
Figure 3.1 Network of Expressways
SU3-17
BUS107
Simulation and Network Modelling
Summary
The world is full of uncertainty, which is what makes simulation so valuable. Simulation
has traditionally not received the attention it deserves in management science courses. The
primary reason for this has been the lack of easy-to-use simulation software. In Chapter
1 of this Study Unit, we have demonstrated the use of the Monte Carlo simulation for
systems of events governed by probability distribution.
In Chapter 2 of this Study Unit, we study physical systems of network. The network
structure of these models allows them to be represented graphically in a way that
is intuitive to users. This graphical representation can then be used as an aid in the
spreadsheet model development. There are many different network models available to
solve specific needs and this chapter has presented three of the more popular network
models.
SU3-18
BUS107
Simulation and Network Modelling
Formative Assessment
1.
The primary difference between simulation models and other types of spreadsheet
models is that simulation models contain ____:
a. deterministic inputs
b. random numbers
c. output cells
d. constraints
2.
Which of the following is not one of the important distinctions of probability
distributions?
a. Discrete versus continuous
b. Symmetric versus skewed
c. Bounded versus unbounded
d. Positive versus negative
3.
Which of the following is typically not an application of simulation models?
a. Operations models
b. Financial models
c. Marketing models
d. Logistics models
4.
Which of the following is the most likely characteristic of a distribution that is to be
used to develop a simulation model for estimating the time until failure of a product
in a simulation model?
a. Unbounded
b. Left skewed
c. Normal
d. Uniform
SU3-19
BUS107
5.
Simulation and Network Modelling
Problems which deal with the direct distribution of products from supply locations
to demand locations are called:
a. transportation problems
b. assignment problems
c. network problems
d. transhipment problems
6.
The objective in transportation problems is typically to:
a. maximize profits
b. maximize revenue
c. minimize costs
d. maximize feasibility
7.
For all routes with positive flows in an optimized transportation problem, the reduced
cost will be:
a. zero
b. how much less shipping costs would have to be for shipments to occur along
that route
c. how much more shipping costs would have to be for shipments to occur along
that route
d. how much the capacity is along that shipping route
8.
The network model representation of a transportation problem has the following
advantage relative to the special case of a simple transportation model:
a. it does not require capacity restrictions on the arcs of the network
b. the flows in the network model don't necessarily have to be from supply
locations to demand locations
c. a network model representation is generally easier to formulate and solve
d. All of these options
SU3-20
BUS107
9.
Simulation and Network Modelling
In a typical network model representation of a transportation problem, the nodes
indicate
a. roads
b. rail lines
c. geographic locations
d. rivers
10. In a typical minimal spanning tree model, the purpose is to determine the
a. shortest distance from starting node to every other node
b. maximum flow from node to node
c. minimum length required for all nodes to be connected
d. fastest time to travel from the start source node to the sink node
11. In a typical shortest-route model, the purpose is to determine the
a. shortest distance from the starting node to every other node
b. maximum flow from node to node
c. minimum length required for all nodes to be connected
d. fastest time to travel from the start source node to the sink node
12. In a typical maximal flow model, the purpose is to determine the
a. shortest distance from the starting node to every other nodes
b. maximum flow from the source node to the sink node
c. minimum length required for all nodes to be connected
d. fastest time to travel from the start source node to the sink node
13. Which of the following is not true for a shortest-route model?
a. The information in the permanent node can always be updated.
b. It is about finding the shortest distance from the starting node to every other
nodes.
SU3-21
BUS107
Simulation and Network Modelling
c. The information in the temporary node can always be updated.
d. It is also known as a shortest-path model.
SU3-22
BUS107
Simulation and Network Modelling
Solutions or Suggested Answers
Chapter 1 Review Questions
1.
What is the Monte Carlo simulation? List the major steps in the Monte Carlo
simulation process.
The concept of the Monte Carlo simulation is experimentation on the probabilistic
elements through random sampling.
The major steps used in the Monte Carlo simulation process are as follows:
1.
Set up a probability distribution and the cumulative probability
distribution of the past performance.
2.
2.
Establish interval of random numbers.
3.
Generate random numbers using the random number tables.
4.
Perform simulation by mapping random numbers with input values.
Simulation has been successfully applied in a variety of applications. List and discuss
two (2) of these areas of applications.
The following are some possible areas of applications:
• New Product Development
• Airline Overbooking
• Inventory Policy
• Traffic Flow
• Waiting Lines
3.
Discuss the advantages and disadvantages of using a simulation approach to a risk
analysis.
Advantages:
SU3-23
BUS107
Simulation and Network Modelling
1.
Simulation provides insights into the problem solutions when other
management science methods fail.
2.
Simulation enables us to project the performance of an existing system
under a proposed set of modifications without disrupting current system
performance. Such performance may be analysed over any time horizon.
3.
Simulation models assist in the design of proposed systems by providing
a convenient experimental laboratory for conducting “what-if” analyses.
Disadvantages:
1.
Simulation models are generally time consuming and expensive to
develop.
2.
Simulation models provide only an estimate of a model’s true parameter
values.
3.
There is no guarantee that the policy shown to be optimal by the
simulation is, in fact, optimal.
4.
Identify the reasons for using simulation as a technique.
These are the possible reasons why simulation is being used in decision-making:
• It can be used for a wide variety of practical problems.
• The simulation approach is relatively easy to explain and understand. As a
result, management confidence is increased and acceptance of the results is
more easily obtained.
• Spreadsheet packages now provide another alternative for model
implementation, and third-party vendors have developed add-ins that
expand the capabilities of the spreadsheet packages.
• Computer software developers have produced simulation packages that
make it easier to develop and implement simulation models for more
complex problems.
SU3-24
BUS107
Simulation and Network Modelling
Chapter 2 Review Questions
1.
Describe the characteristics of the “Shortest-path model” and list three (3) of its
potential applications.
The shortest-path model is to find a path through some of the nodes of the network
which minimises the total distance from the source node to the destination node.
Its usages include:
• Highway travel between cities
• New road construction
• Facility location
2.
Describe the following network models and provide each with Two (2) examples of
their application.
• Transportation
• Assignment
• Traveling Salesman
Model
Description
Transportation
Find the total minimum
cost of shipping goods
from supply points to
destination points.
Applications
• Department store
branch shipments
• Monthly
production
scheduling
• Marketing
strategy
approaches
• Emergency
supply allocation
SU3-25
BUS107
Simulation and Network Modelling
Model
Description
Assignment
Find the minimum cost
assignment of objects to
tasks.
Applications
• Salesmen
to
territories
• Pilots to aircraft
• Programming
tasks
• Machines
to
locations
Travelling Salesman
Find the maximum cost
of visiting all nodes of
a network, returning to
a starting node without
repeating any node.
• Scheduling
service crews
• Designing
robotics
manufacturing
equipment
• Scheduling
security patrols
3.
The fundamental underlying assumption in a transportation model methodology is
that the units of ‘product’ being transported are homogeneous. Discuss how you can
modify the model to achieve better fit.
Modifications to make the appropriate ‘fit’ are:
• Use the ‘typical mix’ or ‘market basket’ approach, where it is assumed that
each truckload carries a standard, homogeneous mix of the firm’s varied
products.
SU3-26
BUS107
Simulation and Network Modelling
• Focus on high volume, high profit items, ignoring in large part the ‘fringe’
products that merely round out the firm’s total product line.
• Observe institutional constraints. It may be, for instance, that leased vehicles
can be used only when the firm’s own truck fleet is fully utilised.
4.
The figure below shows a network of connected expressways with the capacity given
on each section. What is the maximum possible flow between nodes 1 and 7?
Figure 3.1 Network of Expressways
The following solution is just one of several possibilities. However, the total flow
to node 7 should be the same; 23 is the maximum flow.
Iteration
Path
Maximum Flow
1
1→2→5→7
6
2
1→3→6→7
8
3
1→2→4→6→7
7
SU3-27
BUS107
Simulation and Network Modelling
Iteration
Path
Maximum Flow
4
1→3→4→6→7
2
Total:
23
Formative Assessment
1.
The primary difference between simulation models and other types of spreadsheet
models is that simulation models contain ____:
a.
deterministic inputs
Incorrect. Refer to Study Unit 3, Chapter 1.
b.
random numbers
Correct!
c.
output cells
Incorrect. Refer to Study Unit 3, Chapter 1.
d.
constraints
Incorrect. Refer to Study Unit 3, Chapter 1.
2.
Which of the following is not one of the important distinctions of probability
distributions?
a.
Discrete versus continuous
Incorrect. Refer to Study Unit 3, Chapter 1.
b.
Symmetric versus skewed
Incorrect. Refer to Study Unit 3, Chapter 1.
c.
Bounded versus unbounded
Incorrect. Refer to Study Unit 3, Chapter 1.
d.
Positive versus negative
SU3-28
BUS107
Simulation and Network Modelling
Correct!
3.
Which of the following is typically not an application of simulation models?
a.
Operations models
Incorrect. Refer to Study Unit 3, Chapter 1.
b.
Financial models
Incorrect. Refer to Study Unit 3, Chapter 1.
c.
Marketing models
Incorrect. Refer to Study Unit 3, Chapter 1.
d.
Logistics models
Correct!
4.
Which of the following is the most likely characteristic of a distribution that is to be
used to develop a simulation model for estimating the time until failure of a product
in a simulation model?
a.
Unbounded
Incorrect. Refer to Study Unit 3, Chapter 1.
b.
Left skewed
Correct!
c.
Normal
Incorrect. Refer to Study Unit 3, Chapter 1.
d.
Uniform
Incorrect. Refer to Study Unit 3, Chapter 1.
5.
Problems which deal with the direct distribution of products from supply locations
to demand locations are called:
a.
transportation problems
SU3-29
BUS107
Simulation and Network Modelling
Correct!
b.
assignment problems
Incorrect. Refer to Study Unit 3, Chapter 2.
c.
network problems
Incorrect. Refer to Study Unit 3, Chapter 2.
d.
transhipment problems
Incorrect. Refer to Study Unit 3, Chapter 2.
6.
The objective in transportation problems is typically to:
a.
maximize profits
Incorrect. Refer to Study Unit 3, Chapter 2.
b.
maximize revenue
Incorrect. Refer to Study Unit 3, Chapter 2.
c.
minimize costs
Correct!
d.
maximize feasibility
Incorrect. Refer to Study Unit 3, Chapter 2.
7.
For all routes with positive flows in an optimized transportation problem, the reduced
cost will be:
a.
zero
Correct!
b.
how much less shipping costs would have to be for shipments to occur along
that route
Incorrect. Refer to Study Unit 3, Chapter 2.
SU3-30
BUS107
Simulation and Network Modelling
c.
how much more shipping costs would have to be for shipments to occur
along that route
Incorrect. Refer to Study Unit 3, Chapter 2.
d.
how much the capacity is along that shipping route
Incorrect. Refer to Study Unit 3, Chapter 2.
8.
The network model representation of a transportation problem has the following
advantage relative to the special case of a simple transportation model:
a.
it does not require capacity restrictions on the arcs of the network
Incorrect. Refer to Study Unit 3, Chapter 2.
b.
the flows in the network model don't necessarily have to be from supply
locations to demand locations
Correct!
c.
a network model representation is generally easier to formulate and solve
Incorrect. Refer to Study Unit 3, Chapter 2.
d.
All of these options
Incorrect. Refer to Study Unit 3, Chapter 2.
9.
In a typical network model representation of a transportation problem, the nodes
indicate
a.
roads
Incorrect. Refer to Study Unit 3, Chapter 2.
b.
rail lines
Incorrect. Refer to Study Unit 3, Chapter 2.
c.
geographic locations
Correct!
d.
rivers
SU3-31
BUS107
Simulation and Network Modelling
Incorrect. Refer to Study Unit 3, Chapter 2.
10. In a typical minimal spanning tree model, the purpose is to determine the
a.
shortest distance from starting node to every other node
Incorrect. Refer to Study Unit 3, Chapter 2.
b.
maximum flow from node to node
Incorrect. Refer to Study Unit 3, Chapter 2.
c.
minimum length required for all nodes to be connected
Correct!
d.
fastest time to travel from the start source node to the sink node
Incorrect. Refer to Study Unit 3, Chapter 2.
11. In a typical shortest-route model, the purpose is to determine the
a.
shortest distance from the starting node to every other node
Correct!
b.
maximum flow from node to node
Incorrect. Refer to Study Unit 3, Chapter 2.
c.
minimum length required for all nodes to be connected
Incorrect. Refer to Study Unit 3, Chapter 2.
d.
fastest time to travel from the start source node to the sink node
Incorrect. Refer to Study Unit 3, Chapter 2.
12. In a typical maximal flow model, the purpose is to determine the
a.
shortest distance from the starting node to every other nodes
Incorrect. Refer to Study Unit 3, Chapter 2.
b.
maximum flow from the source node to the sink node
SU3-32
BUS107
Simulation and Network Modelling
Correct!
c.
minimum length required for all nodes to be connected
Incorrect. Refer to Study Unit 3, Chapter 2.
d.
fastest time to travel from the start source node to the sink node
Incorrect. Refer to Study Unit 3, Chapter 2.
13. Which of the following is not true for a shortest-route model?
a.
The information in the permanent node can always be updated.
Correct!
b.
It is about finding the shortest distance from the starting node to every other
nodes.
Incorrect. Refer to Study Unit 3, Chapter 2.
c.
The information in the temporary node can always be updated.
Incorrect. Refer to Study Unit 3, Chapter 2.
d.
It is also known as a shortest-path model.
Incorrect. Refer to Study Unit 3, Chapter 2.
SU3-33
BUS107
Simulation and Network Modelling
References
Winston, W. L., & Albright, S. C. (2019). Practical management science (Sixth ed.). Cengage
Learning.
SU3-34