MSE 606 B
Engineering Operations
Research II
Dr. Ahmad R. Sarfaraz
Manufacturing Systems Engineering and Management
California State University, Northridge
Agenda
Course syllabus and administration
Overview of Operations Research II
STANDARD OPERATING PROCEDURES
Collaborative learning groups for research paper and
HW assignments will be utilized
HW assignments and research paper can be worked in a
group of 2-3 students/group
Problems will typically be assigned at each class session
and will form the basis for the examinations
HW assignments will be due at the beginning of the next
class session
One set (the original) should be turned in per group
All students need to have a copy of the HW solution with
them in class
HW is marked as turned in; 5 of the homework
assignments are corrected and graded
Evaluation
Requirement
1.
HW assignments
2.
Exam#1
•
Final
•
Research Paper
Parts
5
1
1
1
Points
30
250
300
300
Total Points
150
250
300
300
Topics Covered
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•
•
•
•
•
•
•
•
•
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Inventory Control (deterministic)
Nonlinear Programming (NLP)
Dynamic Programming (deterministic)
Overview of probability and statistics
Inventory Control (probabilistic)
Forecasting
Decision Analysis
Markov Analysis
Queuing Analysis
Simulation
Game Theory
Organization
First Session
• Introduction of new material and mathematical
development
Second Secession
• Solutions procedures, sample problems, and
applications
The Importance of Inventory
Control
• Why is it so important?
• Total value of all inventory is more than a
$1,000,000,000,000
• More than $4,000 each for every man,
woman, and child in the country
• Reducing a little bit, can enhance
company’s competitiveness
• Exist many models including determinate
and probabilistic models
Nonlinear Programming (NLP)
• Presented
– Linear Programming models and several variations of
the LP models
– Objective functions and the constraints were linear
• Many realistic problems have nonlinear
functions
• When LP problems contain nonlinear functions,
they are referred NLP
• Have a separate name, because they are solved
differently
Dynamic Programming
• An approach for making a sequence of
interrelated decisions
• Applicable to problems that are multistage in
nature
• Example:
– A problem of determining an optimal solution over 1year horizon might be broken into 12 smaller stages
• Decomposes a large problem into a number
smaller problems
• Once all small problems have been solved, we
have optimal solution to large problem
Multicriteria Decision Making:
Analytical Hierarchy Process
• Presented goal programming last semester
• Learned how to formulate a problem with more
than one objectives
• AHP developed by Saati
• A method for rankling decision alternatives and
selecting the best one when the decision maker
has multiple objectives, or criteria
• GP answers “how much?”, whereas AHP
answers “which one?”
Decision Analysis
• In LP formulation, we assumed that certainty
existed
• Means that all of the model coefficients, and
constraint values are known with certainty
• Many decision-making situations occur under
conditions of uncertainty
• Decision situations can be categorized into two
classes: situations in which probabilities can be
assigned to future occurrences and situations in
which probabilities cannot be assigned
• Will present both situations
Markov Analysis
• Like a decision analysis, it is not an optimization
technique
• A probabilistic technique
• Provides probabilistic information about a decision
situation
• Applicable to systems that probabilistic information
moves from one state (condition) to another, over time
• Example:
– Probability that a machine will be running one day breakdown on
the next
– Probability that a customer will change his/her taste from one
month to the next
• Referred to as the “Brand Switching”
Game Theory
• In decision analysis, there is one decision maker
• No competitors whose decisions might change
the decision made by the first one
• Many situations involve several decision makers
who compete with one another to arrive at the
best outcome
• Examples:
– Card games, parlor games, political campaigns,
athletic competitions, military battles, advertising and
marketing campaigns, and so on
Forecasting
• Prediction of what will occur in the future
• Managers are continuously trying to
predict the future
• They usually use judgment, opinion, or
past experiences to forecast
• Mathematical models exist to help
managers
• Will present some of these techniques
Queuing Analysis
• Waiting in queues-waiting lines-is one of the
most occurrences in everyone’s life
• Not only people spend a significant of their time
in lines, but products queue up in production
plants
• Examples: machinery waits to be serviced,
planes wait to take off and land, ships at ports
wait to unload and load, and so on
• Because time is a valuable resource, the
reduction of waiting time is an important topic
Simulation
• Some of the OR topics deal with mathematical
models that can be applied to certain types of
problems
• Not all real-world problems can be solved by
applying a specific type of technique
• When problems cannot be formulated,
simulation is an alternative technique
• Simulation technique can be applied to queuing,
inventory control, production and manufacturing,
finance, marketing, public sector operations, and
environmental and resource analysis
Next Session
• NLP Modeling
– Objective functions
– Decision variables
– Constraints
Inventory Modeling
Why is it Important?
• Pervades the business world
• Necessary for any company dealing with physical
products
– Manufacturing
– Wholesalers
– Retailers
• Total value (in US) is more than
$1000,000,000,000
• 25% associates with storing cost
• Hence, reducing a little bit, can enhance
company’s competitiveness
•
•
Basic Questions in Inventory
Control
How much should we stock?
Two extreme answers to this question:
– A lot
1. This ensures that we never run out
2. An easy way of managing Stock
3. Expensive in inventory costs, cheap in
management costs
– None/very Little
Known as JIT
2. A difficult way of managing stock
3. Cheap in inventory costs, expensive in
management costs
1.
•
When should we order?
Types of Inventory Policies
• Depends on demand and lead time
– the number of units that will need to be
withdrawn from inventory
• Deterministic Models
• Stochastic Models
Types of Inventory Costs
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Purchasing Costs
Holding costs
Ordering costs
Stock out costs
– Not considered here
• Annual Inventory Cost=Purchasing Costs+Holding
Costs+Ordering Costs
Holding Costs
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Storage Costs
Labor
Overheads (Heating, Lighting, Security)
Money Tied up (Loss of Interest, Opportunity
Cost)
• Obsolescence Costs
• Stock Deterioration (Lose Money If Product
Deteriorates)
• Theft/insurance
Ordering Costs
• Clerical/labor Costs of Processing Orders
• Inspection and Return of Poor Quality
Products
• Transport Costs
• Handling Costs
Deterministic Assumptions
•
•
•
•
Demand is known and constant
Lead time is known and constant
Order quantity does not depend on price
Order quantity arrives all at once when
needed
• Planned shortages are not allowed
Basic Model
Q
time
Inventory Control Notation
•
•
•
•
•
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K=ordering cost
c=unit purchasing cost
h=holding cost per unit per unit of time
Q=ordering quantity
a=annual demand
t=cycle time
Annual Holding Cost
•
Annual holding cost = (holding cost per
unit)(Average inventory
•
h(Q/2)
where Q/2 is the average (constant) inventory
level
Annual
Holding
Cost
Holding Cost Curve
Order Quantity
Annual Order Cost
Annual order cost = co(R/Q)
where (R/Q) is the number of orders per year
(R used, Q each order)
Total
Annual
Ordering
Cost
Annual Order Cost
Order Quantity
Total Annual Cost Curve
Q
Optimal Policy
• TC = ch(Q/2) + co(R/Q)
• The function that we want to minimize by
choosing an appropriate value of Q
• Differentiating total cost with respect to Q
and equating to zero :Q* = (2Rco/ch)1/2
• Total annual cost associated with the EOQ
(2Rcoch) 1/2
Assumptions in Deterministic Models
1.
2.
3.
4.
5.
•
•
Demand is known and constant
Lead time is known and constant
Order quantity does not depend on price
Order quantity arrives all at once when needed
Planned shortages are not allowed
Presented EOQ model for a single item
Relaxed the 4th assumption and developed the
EPQ model
EOQ Model with Quantity Discount
• Relax the 3rd assumption
• Quantity discount means
that the order quantity
depends on price
• More quantity at lower
price
• To illustrate the problem,
consider this example
• C1>C2>C3>C4
Price range
Quantity
C1
0-Q1
C2
C3
C4
Q1-Q2
Q2-Q3
>Q3
Graphical Solution: Plot of Cj and Q
TC
C1
C2
C3
Q
Q1
Q2
Q3
Solution Procedure
1. For each unit price, calculate the EOQ
2. If the EOQ is within the feasible range,
calculate the corresponding TC*
3. If the EOQ is not within the feasible
range, calculate TC using the total cost
function
4. Compare the TC for all unit prices and
choose the minimum TC
Example
• Ordering cost:
A=$2500
• Inventory carrying
charge: I=15%
• Annual demand,
D=200 units
• Vender offers the
price discount
Quantity
Price
Holding cost:
H=IC
0-49
$1400
H=1400(0.15)
=$210
50-89
$1100
H=1100(0.15)
=$165
90+
$900
H=900(0.15)
=$135
Solution
•
•
Compute Q* at C1=$1400
Q* = (2DA/IC)1/2=[(2)(2500)(200)/(210)]1/2=69
Quantity
Price
H=IC
– Outside the feasible range
•
Q* = (2DA/IC)1/2=[(2)(2500)(200)/(165)]1/2=78
– Inside the feasible range
•
•
$1400 H=$210
50-89
$1100
H=$165
90+
$900
H=$135
TC*=DC+ (2DAIC)1/2=$232,845
– Must be compared with the TC of lower (lowest
in this particular example) discount price
•
•
0-49
TC=DC+ 2DA/Q+HQ/2
TC=(200)(900)+(2)(200)(2500)/90 +
(135)(90)/2 =$191,630
Since $191,630< $232,845, the maximum
discount price should be taken and 90 units
ordered
The EOQ Model with Shortages
•
Assumptions
1.
2.
3.
4.
Demand is known and constant
Lead time is known and constant
Order quantity does not depend on price
Order quantity arrives all at once when
needed (EPQ case)
5. Planned shortages are allowed
Allowed Shortages or Backordering
• May be worthwhile to permit some shortages to
occur
• Can result savings in holding costs
• Benefit may be offset by the shortage cost
• Sale is not lost; firm does not lose the customer
• Customers wait to have their demand filled from
next order
• Shortage cost is the penalty incurred when we
ran out of stock (often requires expediting and
higher price in shorter lead time)
• All shortages are satisfied from the next order
Graphical Representation of
Backordering
Inventory level
Q-S
Q
time
S
T
t1
t2
Revisiting EOQ Modeling
• Consider only one cycle
• During T (where T=Q/D) one order
(Q) is placed, so the order cost is A
and the purchase cost is QC
• Holding cost is: (Q/2)(H)(T)
• TC for one cycle:
QC+A+(Q/2)(H)(T)
• Annual TC is
{(D/Q)[QC+A+(Q/2)(H)(T)]}
• TC=DC+DA/Q+HQ/2; same thing
we had before
Q
T
Determination of Q and S Values
• Consider just one cycle
• During T (where T=Q/D) one order
(Q) is placed, so the order cost is A
and the purchase cost is QC
• Holding cost is: (Q-S)/2)(H)(t1),
where t1=(Q-S)/D, or (Q-S)2(H/2D)
• If k=shortage cost per unit, shortage
cost/cycle is (K)(S/2)(t2), where t2
=S/D, or KS2/2D
• TC for one cycle:
QC+A+ (Q-S)2(H/2D)+ KS2/2D
• Annual TC
{(D/Q)[QC+A+ (Q-S)2(H/2D)+ KS2/2D]}
TC=DC+DA/Q+ (Q-S)2(QH/2)+ QKS2/2
t2
Q
Q-S
S
t1
Optimal values for Q* and S*
• TC=DC+DA/Q+ (Q-S)2(QH/2)+ QKS2/2
• Partial derivatives of TC with respect to Q
and S are equated to zero
• Q*= (2DA/IC)1/2 ((H+K)/K)1/2
• S*=HQ*/(H+K)
• If K approaches infinity, Q* approaches to
(2DA/IC)1/2
Example