MSE 606 B

Engineering Operations Research II

Dr. Ahmad R. SarfarazManufacturing 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 utilizedHW assignments and research paper can be worked in a group of 2-3 students/groupProblems will typically be assigned at each class session and will form the basis for the examinationsHW assignments will be due at the beginning of the next class sessionOne set (the original) should be turned in per groupAll students need to have a copy of the HW solution with them in classHW is marked as turned in; 5 of the homework assignments are corrected and graded

Evaluation

Requirement Parts Points Total Points

1. HW assignments 5 30 150

2. Exam#1 1 250 250• Final 1 300 300• Research Paper 1 300 300

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

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 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

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 Little1. Known as JIT2. A difficult way of managing stock 3. Cheap in inventory costs, expensive in

management costs• 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• Purchasing Costs• Holding costs • Ordering costs• Stock out costs

– Not considered here• Annual Inventory Cost=Purchasing Costs+Holding

Costs+Ordering Costs

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

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 ModelIn

vent

ory

leve

l

Q

time

Inventory Control Notation• 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

AnnualHolding Cost

Order Quantity

Holding Cost Curve

Annual Order Cost

Annual order cost = co(R/Q) where (R/Q) is the number of orders per year

(R used, Q each order)

TotalAnnualOrdering Cost

Order Quantity

Annual Order Cost

Total Annual Cost Curve

Total A

nnual co

st

Annual holding cost

Annual ordering cost

Cos

t

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. Demand is known and constant

2. Lead time is known and constant

3. Order quantity does not depend on price

4. Order quantity arrives all at once when needed

5. 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 Q1-Q2

C3 Q2-Q3

C4 >Q3

Graphical Solution: Plot of Cj and Q

C1

C2C3

TC

QQ1 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– Outside the feasible range

• Q* = (2DA/IC)1/2=[(2)(2500)(200)/(165)]1/2=78– Inside the feasible range

• TC*=DC+ (2DAIC)1/2=$232,845– Must be compared with the TC of lower (lowest

in this particular example) discount price

• 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

Quantity Price H=IC

0-49 $1400 H=$210

50-89 $1100 H=$165

90+ $900 H=$135

The EOQ Model with Shortages

• Assumptions1. Demand is known and constant

2. Lead time is known and constant

3. Order quantity does not depend on price

4. 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

QQ-S

S

T

t1 t2

time

Inventory level

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

t1

t2 Q

S

Q-S

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