ESM 621 Linear Programming Queuing Theory

1
ESM 621Linear ProgrammingQueuing Theory

• 10 March 2003
• Robert A. Perkins, PE

2
Four Aspects of Engineering Management

• Human Side
• Project Management
• Financial
• Stochastic

3

• designating a process having an infinite
  progression of jointly distributed random
  variables.
• of, pertaining to, or arising from chance
  involving probability random
• from the Greek stochastikos meaning, proceeding
  by guesswork.
•  

4
Operations Research

• Scientific approach to executive problem
  solving
• A collection of techniques that help managers
  make decisions
• AKA Decision science

5
Decision Science

• Misnomer
• Deterministic vs. stochastic
• Operations vs. strategic
• Models

6
Models

• Breadth
• Range of situations for which it applies
• Sensitivity analysis
• Perturbation of of underlying assumptions
• Determines if model is robust

7
Three Rules of Modeling

• What data are relevant
• and how can I find them
• Start with simplest model
• use sensitivity to determine if model should be
  expanded
• No model supplies THE correct answer
• use models interactively for insight that will
  help you bring issues into focus

8
Tonight

• Linear programming
• Optimization
• Efficient use of resources
• Deterministic, for us
• Queuing theory
• Waiting lines
• Stochastic

9
Linear Programming

• Introduction and graphical solutions
• More complex using Excel Solver
• Sensitivity analysis

10
Typical Problem

• Felixco make furniture, specifically tables and
  chairs.
• Each table make a profit of 12 and each chair
  makes a profit of 10.

11
Objective Function

• Maximize Profit, call it Z
• XT number of tables
• XC number of chairs
• Want to maximize Z
• Z 12 XT 10XC
• This is the objective function

12
Constraints

• Felixcos production has several limitations that
  hinder Felixco from maximizing the profit. Three
  processing centers cutting, hammering, and
  painting.

13

• Cutting. Have 330 hours per month available.
  Each table takes 3 hours of cutting each chair
  takes 5 hours.
• Hammering. Have 325 hours per month available.
  Each table takes 5 hours of hammering each chair
  4 hours.
• Painting. Have 150 hours per month. Each table
  3 hours, each chair 1 hour.

14
Constraints

• Cutting 3XT 5XC lt 330 hours
• Hammering 5XT 4XC lt 325 hours
• Painting 3XT 1XC lt 150 hours
• For completeness, XT gt 0 XC gt 0
• And Z 12XT 10XC
• Why not just make all tables?
• Excel

15
Minimization

• Same logic
• Optimal is still at corner
• Desk problem

16
Dietitian wants to minimize total cost of two
types of breakfast foods, while still meeting the
MDR of vitamin A (MDR 16 mg/day) and B (MDR
12 mg/day).  Costs and vitamin content per
portion
17
(No Transcript)
18
Solver

• History
• Excel Solver
• Add-in

19

• Oil Refinery makes fuel oil (FO) and gasoline
  (gas).
• Profit is
• 4 cents per gal for FO and
• 7 cents per gal for gas.
• What is Objective Function?

20
Constraints

• Processing time
• FO, 3 min/gal
• gas, 2 min/gal
• 110 hours per week available
• What is expression for processing time constraint?

21

• Storage
• FO 4 storage units/ gal
• gas take 5 storage units/ gal
• 10,000 storage units available
• What is expression for this constraint.

22
Constraints, cont.

• Emissions
• FO sends 0.5 oz per week
• gas sends 1.0 oz per week
• Allowed 1500 oz per week
• What is expression for the emissions constraint?

23

• From Solver, note that not all resources are
  used.
• Processing time.
• Called slack, processing time is a slack variable.

24
Optimality

• Range of optimality for Objective function
• Range for which changes in objective do not
  change solution.
• Excel

25
How much can I increase the value of fuel oil
from 4 cents and not change the value of the
optimal solution?
26
Shadow Price

• Change in objective function resulting from a 1
  unit increase in right hand side.
• Use Solvers Sensitivity Analysis

27
Shadow price is the same for this range of
constraint values.
28
Queuing Theory

• AKA queueing
• Types of waiting lines
• vehicle unloading
• airport landing
• telecommunications
• production lines
• repair
• emergency room

29
Served customers
customers
The queuing system
Input Source
Queue
Service Mechanism
30
Definitions

• Input source
• potential customers
• AKA calling population
• may be infinite or limited
• Statistical pattern
• Poisson process
• other
• Interarrival time
• time between consecutive arrivals

31

• Queue
• where customers wait before being served
• infinite or finite
• Queue Discipline
• Order customers are served
• FCFS
• LCFS
• Random

32

• Service Mechanism
• How many service channels
• servers
• Service time
• AKA holding time
• Statistical, not equal

33
Notation (Kendall)

• Describe Model
• ____ / ____ / ____ ____ / ____ / ____
• Arrival distribution / Service distribution /
  number of servers
• Queue discipline / Maximum number in queue /
  maximum number in calling population.
• Usually assume FCFS, infinite, and infinite

34
Poisson

• Describes the probability distribution of
  independent events
• The random variable is the number of times an
  event occurs in a single unit of time.
• The average number of occurrences of events
  remains constant
• If we have 3 accidents per week.
• Number of accidents in the random variable
• Assume that is the same week in week out.
• Then it can be described by a Poisson
  distribution.
• With ? (mu) the mean or average occurrence.

35
If there are 3 accidents in an average week, what
is the probability there will be 5 accidents in a
week?
P(5 3)
36

• M / M / 1
• Poisson arrivals, Poisson service, and one server.

37
M / M / 1

• ? arrival rate (Poisson from calling
  population)
• ? service rate (Poisson)
• k 1
• FCFS
• Infinite queue length possible
• Infinite calling population

38
M / M / 1 Formulae

• Need ? / ? lt k or less than 1
• That is, service rate must be greater than
  arrival rate.

39
(No Transcript)
40
(No Transcript)
41
(No Transcript)
42
Utilization factor

• The ? / ? is called the utilization factor
• It is the probability the server is busy
• Mean arrival rate divided by the mean service
  rate.

43
Schipps Trucking

• 3 trucks arrive per hour
• Schipps can unload 4 trucks per hours.
• Excel

44
Cost of Service
Cost of Waiting

Cost of Service
Level of Service
45
Where to from here?

• Multiple channels
• Distributions other than Poisson
• Limited calling population
• Discipline other than FCFS

46
Where to from here?