Transcript Document 4118275

NEW MEXICO INSTITUTE OF
MINING AND TECHNOLOGY
Department of Management
Management Science for Engineering
Management (EMGT 501)
Instructor
: Toshi Sueyoshi (Ph.D.)
HP address
: www.nmt.edu/~toshi
E-mail Address : [email protected]
Office
: Speare 143-A
1. Course Description:
The purpose of this course is to introduce
Management Science (MS) techniques for
manufacturing, services, and public sector.
MS includes a variety of techniques used in modeling
business applications for both better understanding
the system in question and making best decisions.
MS techniques have been applied in many
situations, ranging from inventory management
in manufacturing firms to capital budgeting in
large and small organizations.
Public and Private Sector Applications
The main objective of this graduate course is to
provide engineers with a variety of decisional tools
available for modeling and solving problems in a
real business and/or nonprofit context.
In this class, each individual will explore how to
make various business models and how to solve
them effectively.
2. Texts -- The texts for this course:
(1) Anderson, Sweeney, Williams & Martin
An Introduction to Management Science:
Quantitative Approaches to Decision Making,
Thomson South-Western (Required)
3. Grading:
In a course, like this class, homework problems are
essential. We will have homework assignments.
Homework has significant weight. The grade
allocation is separated as follows:
Homework
Mid-Term Exam
Final Exam
20%
40%
40%
The usual scale (90-100=A, 80-89.99=B, 7079.99=C, 60-69.99=D) will be used.
Please remember no makeup exam.
4. Course Outline:
Week
1
2
3
4
5
6
7
8
Topic(s)
Introduction and Overview
Linear Programming
Solving LP and Dual
DEA
Game Theory
Project Scheduling: PERT-CPM
Inventory Models
Review for Mid-Term EXAM
Text(s)
Ch. 1&2
Ch. 3&17
Ch. 4&18
Ch. 5
Ch.5
Ch. 9
Ch. 10
Week
9
10
11
12
13
14
15
16
Topic(s)
Text(s)
Waiting Line Models Ch. 11
Waiting Line Models Ch. 11
Decision Analysis
Ch. 13
Multi-criteria Decision Ch. 14
Forecasting
Ch. 15
Markov Process
Ch. 16
Slack (for Class Delay)
Review for FINAL EXAM
Assessment
• Please indicate the current level of your knowledge. (1: no
idea, 2: little, 3: considerable, 4: very well).
•
•
•
•
•
•
•
Topic
(1) Linear Programming
(2) Dual and Primal Relationship
(3) Simplex Method
(4) Data Envelopment Analysis
(5) PERT/CPM
(6) Inventory
Your Assessment
• Return the assessment by Sep 1 (noon) to [email protected]
Model Development
• Models are representations of real objects or
situations
– Mathematical models - represent real
world problems through a system of
mathematical formulas and expressions
based on key assumptions, estimates, or
statistical analyses
Advantages of Models
• Generally, experimenting with models
(compared to experimenting with the real
situation):
– requires less time
– is less expensive
– involves less risk
Mathematical Models
• Cost/benefit considerations must be made in
selecting an appropriate mathematical model.
• Frequently a less complicated (and perhaps less
precise) model is more appropriate than a more
complex and accurate one due to cost and ease of
solution considerations.
Mathematical Models
• Relate decision variables (controllable inputs) with fixed or
variable parameters (uncontrollable inputs)
• Frequently seek to maximize or minimize some objective
function subject to constraints
• Are said to be stochastic if any of the uncontrollable inputs is
subject to variation, otherwise are deterministic
• Generally, stochastic models are more difficult to analyze.
• The values of the decision variables that provide the
mathematically-best output are referred to as the optimal
solution for the model.
Body of Knowledge
• The body of knowledge involving
quantitative approaches to decision making is
referred to as
– Management Science
– Operations research
– Decision science
• It had its early roots in World War II and is
flourishing in business and industry with the
aid of computers
Transforming Model Inputs into
OutputInputs
Uncontrollable
(Environmental Factors)
Controllable
Inputs
(Decision
Variables)
Mathematical
Model
Output
(Projected
Results)
Example: Project Scheduling
Consider the construction of a 250-unit apartment
complex. The project consists of hundreds of
activities involving excavating,
framing, wiring, plastering, painting,
land-scaping, and more.
Some of the activities must be done
sequentially and others can be done
at the same time. Also, some of the activities can be
completed faster than normal by purchasing
additional resources (workers, equipment, etc.).
Example: Project Scheduling
• Question: What is the best schedule for the
activities and for which activities should additional
resources be purchased? How could management
science be used to solve this problem?
• Answer: Management science can provide a
structured, quantitative approach for determining
the minimum project completion time based on the
activities' normal times and then based on the
activities' expedited (reduced) times.
Example: Project Scheduling
• Question: What would be the decision variables of the
mathematical model? The objective function? The
constraints?
• Answer:
– Decision variables: which activities to expedite and
by how much, and when to start each activity
– Objective function: minimize project completion
time
– Constraints: do not violate any activity precedence
relationships and do not expedite in excess of the
funds available.
Example: Project Scheduling
• Question:
Is the model deterministic or stochastic?
• Answer:
Stochastic. Activity completion times, both
normal and expedited, are uncertain and subject to
variation. Activity expediting costs are uncertain.
The number of activities and their precedence
relationships might change before the project is
completed due to a project design change.
Example: Project Scheduling
• Question:
Suggest assumptions that could be made to
simplify the model.
• Answer:
Make the model deterministic by assuming normal
and expedited activity times are known with
certainty and are constant. The same assumption
might be made about the other stochastic,
uncontrollable inputs.
Data Preparation
• Data preparation is not a trivial step, due to
the time required and the possibility of data
collection errors.
• A model with 50 decision variables and 25
constraints could have over 1300 data
elements!
• Often, a fairly large data base is needed.
• Information systems specialists might be
needed.
Model Solution
• The “best” output is the optimal solution.
• If the alternative does not satisfy all of the
model constraints, it is rejected as being
infeasible, regardless of the objective
function value.
• If the alternative satisfies all of the model
constraints, it is feasible and a candidate for
the “best” solution.
Computer Software
• A variety of software packages are available
for solving mathematical models.
• a) Management Scientist Software (attached
to the text book)
• b) QSB and Spreadsheet packages such as
Microsoft Excel
Model Testing and Validation
• Often, goodness/accuracy of a model cannot be assessed
until solutions are generated.
• Small test problems having known, or at least expected,
solutions can be used for model testing and validation.
• If the model generates expected solutions, use the model on
the full-scale problem.
• If inaccuracies or potential shortcomings inherent in the
model are identified, take corrective action such as:
– Collection of more-accurate input data
– Modification of the model
Report Generation
• A managerial report, based on the results of the model,
should be prepared.
• The report should be easily understood by the decision
maker.
• The report should include:
– the recommended decision
– other pertinent information about the results (for
example, how sensitive the model solution is to the
assumptions and data used in the model)
Implementation and Follow-Up
• Successful implementation of model results is of
critical importance.
• Secure as much user involvement as possible
throughout the modeling process.
• Continue to monitor the contribution of the model.
• It might be necessary to refine or expand the
model.
Linear Programming (LP):
A mathematical method that consists of an objective
function and many constraints.
LP involves the planning of activities to obtain an
optimal result, using a mathematical model, in which
all the functions are expressed by a linear relation.
A standard Linear Programming Problem
Maximize 3 x1  5 x2
subject to 1x1  0 x2  4
0 x1  2 x2  12
3 x1  2 x2  18
x1  0, x2  0
Applications: Man Power Design, Portfolio Analysis
Simplex method:
A remarkably efficient solution procedure for
solving various LP problems.
Extensions and variations of the simplex method
are used to perform postoptimality analysis
(including sensitivity analysis).
(a) Algebraic Form
(0) Z  3x1  5x2
x1
 x3
(1)
x2
(2)
3x1  2 x2
(3)
(b) Tabular Form
Basic Variable Eq.
Z
x3
x4
x5
(0)
(1)
(2)
(3)
 x4
 x5
0
4
 12
 18
Coefficient of:
Z
1
0
0
0
x1 x2 x3 x4 x5
-3
1
2
3
-5
0
0
2
0 0
1 0
0 1
0 0
0
0
0
1
Right
Side
0
0
12
18
Duality Theory:
An important discovery in the early development
of LP is Duality Theory.
Each LP problem, referred to as ” a primal
problem” is associated with another LP problem
called “a dual problem”.
One of the key uses of duality theory lies in the
interpretation and implementation of sensitivity
analysis.
PERT (Program Evaluation and Review
Technique)-CPM (Critical Path Method):
PERT and CPM have been used extensively to
assist project managers in planning, scheduling,
and controlling their projects.
Applications: Project Management, Project
Scheduling
START 0
Critical Path
A 2
2 + 4 + 10 + 4 + 5 + 8
+ 5 + 6 = 44 weeks
B 4
10
C
D 6
E 4
G 7
H 9
I 7
F 5
J 8
L 5
K 4
M 2
N 6
FINISH 0
Decision Analysis:
An important technique for decision making in
uncertainty.
It divides decision making between the cases
of without experimentation and with
experimentation.
Applications: Decision Making, Planning
decision fork
chance fork
f
c
b
g
d
a
h
e
Markov Chain Model:
A special kind of a stochastic process.
It has a special property that probabilities,
involving how a process will evolve in
future, depend only on the present state of
the process, and so are independent of events
in the past.
Applications: Inventory Control, Forecasting
Queueing Theory:
This theory studies queueing systems by
formulating mathematical models of their
operation and then using these models to derive
measures of performance.
This analysis provides vital information for
effectively designing queueing systems that
achieve an appropriate balance between the
cost of providing a service and the cost
associated with waiting for the service.
Served customers
Queueing system
C
C
CCCCCC
C
C
Queue
Customers
S
S
S
S
Service
facility
Served customers
Applications: Waiting Line Design, Banking,
Network Design
Inventory Theory:
This theory is used by both wholesalers and retailers
to maintain inventories of goods to be available for
purchase by customers.
The just-in-time inventory system is such an example
that emphasizes planning and scheduling so that the
needed materials arrive “just-in-time” for their use.
Applications: Inventory Analysis, Warehouse Design
Economic Order Quantity (EOQ) model
Inventory
level
Q
Batch
size Q
0
Q  at
Q
a
2Q
a
Time t
Forecasting:
When historical sales data are available, statistical
forecasting methods have been developed for using
these data to forecast future demand.
Several judgmental forecasting methods use expert
judgment.
Applications: Future Prediction, Inventory Analysis
Monthly sales (units sold)
The evolution of the monthly sales of a product
illustrates a time series
10,000
8,000
6,000
4,000
2,000
0
1/99 4/99 7/99 10/99 1/00 4/00 7/00
Introduction to MS/OR
MS: Management Science
OR: Operations Research
Key components: (a) Modeling/Formulation
(b) Algorithm
(c) Application
Management Science (MS)
(1) A discipline that attempts to aid managerial
decision making by applying a scientific approach
to managerial problems that involve quantitative
factors.
(2) MS is based upon mathematics, computer
science and other social sciences like economics
and business.
General Steps of MS
Step 1: Define problem and gather data
Step 2: Formulate a mathematical model to
represent the problem
Step 3: Develop a computer based procedure
for deriving a solution(s) to the
problem
Step 4: Test the model and refine it as needed
Step 5: Apply the model to analyze the
problem and make recommendation
for management
Step 6: Help implementation
Linear Programming (LP)
Linear Programming (LP) Problem
• The maximization or minimization of some quantity
is the objective in all linear programming problems.
• All LP problems have constraints that limit the
degree to which the objective can be pursued.
• A feasible solution satisfies all the problem's
constraints.
• An optimal solution is a feasible solution that results
in the largest possible objective function value when
maximizing (or smallest when minimizing).
• A graphical solution method can be used to
solve a linear program with two variables.
Linear Programming (LP) Problem
• If both the objective function and the constraints
are linear, the problem is referred to as a linear
programming problem.
• Linear functions are functions in which each
variable appears in a separate term raised to the
first power and is multiplied by a constant (which
could be 0).
• Linear constraints are linear functions that are
restricted to be "less than or equal to", "equal to",
or "greater than or equal to" a constant.
Problem Formulation
• Problem formulation or modeling is the
process of translating a verbal statement of
a problem into a mathematical statement.
[1] LP Formulation
(a) Decision Variables : x1 , x2 ,, xn
All the decision variables are non-negative.
(b) Objective Function : Min or Max
(c) Constraints
Minimize
s.t.
2 x1  3x2
3x1  4 x2  3
4 x1  1x2  4
s.t. : subject to
x1  0, x2  0
Guidelines for Model
Formulation
•
•
•
•
•
Understand the problem thoroughly.
Describe the objective.
Describe each constraint.
Define the decision variables.
Write the objective in terms of the decision
variables.
• Write the constraints in terms of the
decision variables.
[2] Example
A company has three plants, Plant 1, Plant 2, Plant
3. Because of declining earnings, top management
has decided to revamp the company’s product line.
Product 1: It requires some of production capacity
in Plants 1 and 3.
Product 2: It needs Plants 2 and 3.
The marketing division has concluded that the
company could sell as much as could be
produced by these plants.
However, because both products would be
competing for the same production capacity in
Plant 3, it is not clear which mix of the two
products would be most profitable.
The data needed to be gathered:
1. Number of hours of production time available
per week in each plant for these new products.
(The available capacity for the new products is
quite limited.)
2. Production time used in each plant for each
batch to yield each new product.
3. There is a profit per batch from a new
product.
Production Time
per Batch, Hours
Plant
1
1
1
2
0
Production Time
Available
per Week, Hours
4
2
0
2
12
2
18
Product
3
3
Profit
$3,000
per batch
$5,000
x1 : # of batches of product 1 produced per week
x2: # of batches of product 2 produced per week
Z : the total profit per week
Maximize 3x1  5x 2
subject to 1x  0 x  4
1
2
0 x1  2 x 2  12
3x1  2 x 2  18
x1  0, x 2  0
Graphic Solution
x1  0, x2  0
x2
10
8
6
4
Feasible
region
2
0
2
4
6
8
x1
x1  0, x2  0
x2
10
8
x1  4
6
4
Feasible
region
2
0
2
4
6
8
x1
x1  0, x2  0
x2
10
8
x1  4
6
2 x2  12
4
Feasible
region
2
0
2
4
6
8
x1
x1  0, x2  0
x2
10
8
3x1  2 x2  18
x1  4
6
2 x2  12
4
Feasible
region
2
0
2
4
6
8
x1
Maximize: 3x1  5 x2
Slope-intercept form: Z  3x1  5x2
3
1 

 x2   x1  Z 
5
5 

The largest value x
2
Z  36  3x1  5x2 8
The optimal solution
( 2,6)
6
Z  20  3x1  5x2
Z  10  3x1  5x2
4
2
0
2
4
6
8
10
x1
Summary of the Graphical Solution Procedure
for Maximization Problems
• Prepare a graph of the feasible solutions for each
of the constraints.
• Determine the feasible region that satisfies all the
constraints simultaneously..
• Draw an objective function line.
• Move parallel objective function lines toward
larger objective function values without entirely
leaving the feasible region.
• Any feasible solution on the objective function
line with the largest value is an optimal solution.
[4] Standard Form of LP Model
Maximize
c1x1  c2x 2    cn x n
s.t.
a11 x1  a12 x2    a1n xn  b1
a21 x1  a22 x2    a2 n xn  b2



am1 x1  am 2 x2    amn xn  bm
x1  0, x2  0, , xn  0
[5] Other Forms
The other LP forms are the following:
1. Minimizing the objective function:
Minimize
Z  c1x1  c2 x2   cn xn .
2. Greater-than-or-equal-to constraints:
a i1x1  a i2x 2    a in x n  bi
3. Some functional constraints in equation form:
ai1x1  ai 2 x2   ain xn  bi
4. Deleting the nonnegativity constraints for
some decision variables:

x j : unrestricted in sign x j  xpj  xn j

[6] Key Terminology
(a) A feasible solution is a solution
for which all constraints are satisfied
(b) An infeasible solution is a solution
for which at least one constraint is violated
(c) A feasible region is a collection
of all feasible solutions
(d) An optimal solution is a feasible solution
that has the most favorable value of
the objective function
(e) Multiple optimal solutions have an infinite
number of solutions with the same
optimal objective value
Multiple optimal solutions:
Example
Maximize
Subject to
and
Z  3x1  2 x2 ,
4
x1
2 x2  12
3x1  2 x2  18
x1  0, x2  0
x2
Z  18  3x1  2 x2
8
Multiple optimal solutions
6
Every point on this red line
segment is optimal,
each with Z=18.
4
Feasible
2 region
0
2
4
6
8 10
x1
(f) An unbounded solution occurs when
the constraints do not prevent improving
the value of the objective function.
x2
x1
Case Study - Personal Scheduling
UNION AIRWAYS needs to hire additional
customer service agents.
Management recognizes the need for cost control
while also consistently providing a satisfactory
level of service to customers.
Based on the new schedule of flights, an analysis
has been made of the minimum number of
customer service agents that need to be on duty at
different times of the day to provide a satisfactory
level of service.
Time Period Covered Minimum #
of Agents
Shift
Time Period
1 2 3 4 5 needed
48
6:00 am to 8:00 am *
79
8:00 am to10:00 am * *
65
10:00 am to noon
* *
87
Noon to 2:00 pm
* * *
64
2:00 pm to 4:00 pm
* *
73
4:00 pm to 6:00 pm
* *
82
6:00 pm to 8:00 pm
* *
43
8:00 pm to 10:00 pm
*
52
10:00 pm to midnight
* *
15
Midnight to 6:00 am
*
Daily cost per agent 170 160 175 180 195
The problem is to determine how many agents
should be assigned to the respective shifts each
day to minimize the total personnel cost for
agents, while meeting (or surpassing) the service
requirements.
Activities correspond to shifts, where the level of
each activity is the number of agents assigned to
that shift.
This problem involves finding the best mix of
shift sizes.
x1 : # of agents for shift 1 (6AM - 2PM)
x2 : # of agents for shift 2 (8AM - 4PM)
x3 : # of agents for shift 3 (Noon - 8PM)
x4 : # of agents for shift 4 (4PM - Midnight)
x5 : # of agents for shift 5 (10PM - 6AM)
The objective is to minimize the total cost of
the agents assigned to the five shifts.
Min 170x1  160x2  175x3  180x4  195x5
x1
 48
s.t.
all xi  0
x1  x2
 79
x1  x2
 65
x1  x2  x3
 87
x2  x3
 64
(i  1 ~ 5)
x3  x4
 73
x3  x4
 82
x4
 43
x4  x5  5 2
x5  1 5
x1
 48
x1  x2
 79
x1  x2  x3
 87
x2  x3
 64
x3  x4
 82
x4
 43
x4  x5  52
x5  15

1

2

3

4

5
( x , x , x , x , x )  (48,31,39,43,15)
Total Personal Cost = $30,610
Slack and Surplus Variables
• A linear program in which all the variables are nonnegative and all the constraints are equalities is said to be
in standard form.
• Standard form is attained by adding slack variables to "less
than or equal to" constraints, and by subtracting surplus
variables from "greater than or equal to" constraints.
• Slack and surplus variables represent the difference
between the left and right sides of the constraints.
• Slack and surplus variables have objective function
coefficients equal to 0.
Example 1: Standard Form
Max
s.t.
5x1 + 7x2 + 0s1 + 0s2 + 0s3
x1
+ s1
= 6
2x1 + 3x2
+ s2
= 19
x1 + x2
+ s3 = 8
x1 , x2 , s1 , s2 , s3 > 0
Interpretation of Computer Output
• In this chapter we will discuss the following output:
– objective function value
– values of the decision variables
– reduced costs
– slack/surplus
• In the next chapter we will discuss how an optimal solution
is affected by a change in:
– a coefficient of the objective function
– the right-hand side value of a constraint
Example 1: Spreadsheet Solution
A
B
C
Optimal Decision Variable Values
X1
X2
5.0
3.0
D
• Partial Spreadsheet Showing Solution
8
9
10
11
12
13
14
15
16
17
Maximized Objective Function
Constraints
#1
#2
#3
Amount Used
5
19
8
46.0
<=
<=
<=
RHS Limits
6
19
8
Example 1: Spreadsheet Solution
Adjustable Cells
Final Reduced Objective
Cell Name Value
Cost
Coefficient
$B$8
X1
5.0
0.0
5
$C$8
X2
3.0
0.0
7
• Reduced Costs
Allowable
Increase
Allowable
Decrease
2 0.333333333
0.5
2
Constraints
Final Shadow Constraint
Allowable
Allowable
Cell Name Value
Price
R.H. Side
Increase
Decrease
$B$13 #1
5
0
6
1E+30
1
$B$14 #2
19
2
19
5
1
$B$15 #3
8
1
8 0.333333333 1.666666667