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Lecture 7 Models ISE 222 Spring 2005 Notes & Course Materials engr.sjsu/kcorker [email protected]

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### Lecture 7 Models ISE 222 Spring 2005Notes & Course Materials www.engr.sjsu.edu/[email protected]

Kevin Corker

San Jose State University

3/5/05

Agenda

- Decision Making
- Under uncertainty and stress
- Econometric Models
- Future Value
- Optimization
- Queuing Models
- CPM & PERT

Models

- Model is a system of relationships whose structure represents some critical dimensions of the real world
- Models are incomplete & Simplifications
- Models – at some point have the finger print of the modeler on them.

Decision Making

- Require Common Metrics Among Alternatives
- Models can be either normative (this is the way things should be) or descriptive (this is the way things are)
- Models support indirect experimentation
- Symbols or representation of things – rather than the things themselves are manipulated.

Common MetricsMeeting the requirement

- “I speak the language of a race whose apogee of intelligence is contained in the phrase ‘time is money’”
- Stephan Daedelus: James Joyce Ulysses

Money Flow Modeling

- Equivalence
- Present equivalence
- Annual equivalence
- Future equivalence

Equivalence = f(Ft, i,n)

where:

Ft is the money flow at the end of time t

i is the interest rate over time t

n is the number of such t increments

Evaluation by economic optimization

- Includes, investment costs, periodic costs, project lifecycle costs
- Need a model
- F= f(X,Y)

Where x represents controllable and y represents uncontrollable system variables

Or

- F = f(X, Y(d), Y(i))

Where Y(d) is associated with design decisions and Y(i) is associated with design independent factors

Decision Evaluation Matrix

- Examination of one of several outcomes depending on which of several futures occurs
- Links a finite set of outcomes to a finite set of futures (and their associated probabilities)
- For those looking for continuity… this is very much the same construct as associated with risk assessment in which the risk is a function of the probability of an event and its consequence

Uncertainty (cont)

- Laplace Criterion : Assume equal likelihood
- Maximin & maximax : extreme criterion
- Hurwicz rule

Maxi {a [max E(ij)] + (1-a) [minj E(ij)]}

Effectiveness

- Single Criterion is not the usual case
- Effectiveness is mission fulfillment measured against multiple criteria
- One of those is the time value of money

Time value of money

- Interest: fee for use of money
- Compound interest: interest earned at the end of the interest period is either paid or earns interest on its itself (i.e. is added to the capital)
- See hand outs

Break Even Calculations

- Relates fixed and variable costs to some measure of operation (duty cycle)
- Buy COTS
- Self Manufacture

TCm = init costs + manufact costs * n

TCb = cost*n

Break even

TCm = TCb

- Lease or Buy

Rate of Return

- Interest rate that causes the equivalent receipts of a money flow to be equal to the equivalent disbursement of that money flow

0 = PE(i*) = S F(t)*(1+ (i*) –t

Solve for the interest rate

- Payout Evaluation
- Amount of tine for the present value of receipts to equal the present value of disbursements.

Optimization

- One decision variable: Differential calculus methods to find the instantaneous rate of change
- Multiple Decision variables: partial differentiation with respect to selected independent variables
- Unconstrained optimization
- f(x) optimum point has to have first derivative of x* is zero and
- f(x) second derivative of x* is either negative (minimum) or positive (maximum)

Bay Bridge example

- Initial Investment Costs :
- Superstructure vs. Piers

E = f(x.y)

- E = eval of total first costs
- X = design variable ein the span between piers (s)
- Y – system parameter of bridge length (l), superstructure weight (w), erected cost of super structure/lb (Cs), and cost of pier/pier (Cp)
- TFC is the total fixed cost if the bridge

d(TFC)/dS = 0

Queuing Theory

- Queuing Theory Queuing Theory examines the progress of customers pursuing offered services.
- 3 areas of focus: Arrivals, Queue, Service Facility - each of which is further subdivided by a variety of analytic detail.
- A single process may consist of more than one stage/station, if the customer passes through a series of Service Facilities;
- It may also have more than one actual Queue/channel/ waiting-line leading to the subsequent Service Facility(ies).

Processes

- arrival process:
- how customers arrive e.g. singly or in groups (batch or bulk arrivals)
- how the arrivals are distributed in time (e.g. what is the probability distribution of time between successive arrivals (the interarrival time distribution))
- Interarrival time distribution:
- regular arrivals (i.e. the same constant time interval between successive arrivals).
- Poisson stream of arrivals corresponds to arrivals at random. In a Poisson stream successive customers arrive after intervals which independently are exponentially distributed.
- average arrival rate single defining parameter of Poisson Distribution.
- scheduled arrivals; batch arrivals; and time dependent arrival rates (i.e. the arrival rate varies according to the time of day).

Service Mechanism

- a description of the resources needed for service to begin
- how long the service will take (the service time distribution)
- the number of servers available
- whether the servers are in series (each server has a separate queue) or in parallel (one queue for all servers)
- whether preemption is allowed (a server can stop processing a customer to deal with another "emergency" customer)

Queue Characteristics:

- how, from the set of customers waiting for service, do we choose the one to be served next (e.g. FIFO (first-in first-out) - also known as FCFS (first-come first served); LIFO (last-in first-out); randomly) (this is often called the queue discipline)
- do we have:
- balking (customers deciding not to join the queue if it is too long)
- reneging (customers leave the queue if they have waited too long for service)
- jockeying (customers switch between queues if they think they will get served faster by so doing)
- a queue of finite capacity or (effectively) of infinite capacity

Questions

- How long does a customer expect to wait in the queue before they are served, and how long will they have to wait before the service is complete?
- What is the probability of a customer having to wait longer than a given time interval before they are served?
- What is the average length of the queue?
- What is the probability that the queue will exceed a certain length?
- What is the expected utilization of the server and the expected time period during which he will be fully occupied (remember servers cost us money so we need to keep them busy).
- In fact if we can assign costs to factors such as customer waiting time and server idle time then we can investigate how to design a system at minimum total cost.
- Is it worthwhile to invest effort in reducing the service time?
- How many servers should be employed?
- Should priorities for certain types of customers be introduced?
- Is the waiting area for customers adequate?

Is it worthwhile to invest effort in reducing the service time?

- How many servers should be employed?
- Should priorities for certain types of customers be introduced?
- Is the waiting area for customers adequate?

Strategies / Heuristics

- Dominance (not good independent of future)
- Aspiration Level (minimax)
- Most Probable Future
- Expected Value

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