MSE 606 B Engineering Operations Research II

1
MSE 606 BEngineering Operations Research II

• Dr. Ahmad R. SarfarazManufacturing Systems
  Engineering and Management
• California State University, Northridge

2
Agenda

• Course syllabus and administration
• Overview of Operations Research II

3
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

4
Evaluation

• Requirement Parts Points Total Points
• HW assignments 5 30 150
• Exam1 1 250 250
• Final 1 300 300
• Research Paper 1 300 300

5
Topics Covered

• 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

6
Organization

• First Session
• Introduction of new material and mathematical
  development
• Second Secession
• Solutions procedures, sample problems, and
  applications

7
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 companys
  competitiveness
• Exist many models including determinate and
  probabilistic models

8
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

9
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
  1-year 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

10
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?

11
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

12
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

13
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

14
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

15
Queuing Analysis

• Waiting in queues-waiting lines-is one of the
  most occurrences in everyones 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

16
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

17
Next Session

• NLP Modeling
• Objective functions
• Decision variables
• Constraints

18
Inventory Modeling
19
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
  companys competitiveness

20
Basic Questions in Inventory Control

• How much should we stock?
• Two extreme answers to this question
• A lot
• This ensures that we never run out
• An easy way of managing Stock
• Expensive in inventory costs, cheap in
  management costs
• None/very Little
• Known as JIT
• A difficult way of managing stock
• Cheap in inventory costs, expensive in
  management costs
• When should we order?

21
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

22
Types of Inventory Costs

• Purchasing Costs
• Holding costs
• Ordering costs
• Stock out costs
• Not considered here
• Annual Inventory CostPurchasing CostsHolding
  CostsOrdering Costs

23
Holding Costs

• 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

24
Ordering Costs

• Clerical/labor Costs of Processing Orders
• Inspection and Return of Poor Quality Products
• Transport Costs
• Handling Costs

25
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

26
Basic Model
Q
Inventory level
time
27
Inventory Control Notation

• Kordering cost
• cunit purchasing cost
• hholding cost per unit per unit of time
• Qordering quantity
• aannual demand
• tcycle time

28
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
29
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
30
Total Annual Cost Curve
Total Annual cost
Cost
Annual holding cost
Annual ordering cost
Q
31
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

32
Assumptions in Deterministic Models

• 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

33
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
• C1gtC2gtC3gtC4

34
Graphical Solution Plot of Cj and Q
TC
C1
C2
C3
Q
Q1
Q2
Q3
35
Solution Procedure

• For each unit price, calculate the EOQ
• If the EOQ is within the feasible range,
  calculate the corresponding TC
• If the EOQ is not within the feasible range,
  calculate TC using the total cost function
• Compare the TC for all unit prices and choose the
  minimum TC

36
Example

• Ordering cost A2500
• Inventory carrying charge I15
• Annual demand, D200 units
• Vender offers the price discount

37
Solution

• Compute Q at C11400
• Q (2DA/IC)1/2(2)(2500)(200)/(210)1/269
• Outside the feasible range
• Q (2DA/IC)1/2(2)(2500)(200)/(165)1/278
• Inside the feasible range
• TCDC (2DAIC)1/2232,845
• Must be compared with the TC of lower (lowest in
  this particular example) discount price
• TCDC 2DA/QHQ/2
• TC(200)(900)(2)(200)(2500)/90 (135)(90)/2
  191,630
• Since 191,630lt 232,845, the maximum discount
  price should be taken and 90 units ordered

38
The EOQ Model with Shortages

• 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
  (EPQ case)
• Planned shortages are allowed

39
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

40
Graphical Representation of Backordering
Inventory level
Q-S
Q
time
S
T
t1
t2
41
Revisiting EOQ Modeling

• Consider only one cycle
• During T (where TQ/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 QCA(Q/2)(H)(T)
• Annual TC is (D/Q)QCA(Q/2)(H)(T)
• TCDCDA/QHQ/2 same thing we had before

Q
T
42
Determination of Q and S Values

• Consider just one cycle
• During T (where TQ/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 kshortage cost per unit, shortage cost/cycle
  is (K)(S/2)(t2), where t2 S/D, or KS2/2D
• TC for one cycle
• QCA (Q-S)2(H/2D) KS2/2D
• Annual TC
• (D/Q)QCA (Q-S)2(H/2D) KS2/2D
• TCDCDA/Q (Q-S)2(QH/2) QKS2/2

t2
Q
Q-S
S
t1
43
Optimal values for Q and S

• TCDCDA/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 ((HK)/K)1/2
• SHQ/(HK)
• If K approaches infinity, Q approaches to
  (2DA/IC)1/2

44
Example