Modelling Flashcards
Irreducable
A chain is irreducible if every state can be reached by every other state
Closed state
If the system once in one of the states of the set will then remain there indefinitely
Absorbing state
Closed set with only one state
N from fundamental matrix
N = ( I - Q )^-1
The expected number of times the process is in the transient
Balking
Behaviour where potential customers or clients decide not to join the queue because it is too long or moving too slowly
Renegading
Behaviour where customers who have already joined a queue decide to leave it before receiving the service
Jockeying
For where there are more than one channels,
where a customer switches from one line to another in attempt to find a faster queue
Waiting lines total costs
Waiting costs - Decrease as speed of service rises
Service cost - Increase as speed of service rises
How to recognise Poisson distribution? (2 crude rules)
- Random arrivals
- Sample mean and variance will be approximately equal
Assumptions of waiting line models (7)
- First in, first out
- All customers wait regardless of queue length
- arrivals independent of preceding arrivals, but avg arrivals dont change over time
- Infinite population arrive by poisson dist
- Service times vary and are independent, but Avg is known
- Service times are negative exponential probability distribution
- Average service rate is greater than average arrival rate
Total costs model for waiting lines
CwLs + CsK
Arbitrary service times
Service times that do not follow a specific distribution pattern.
Reflective of reality
Reasons for simulation (4)
- Actual environment too hard to observe
- not possible to develop analytical solution
- not sufficient time to allow time to operate extensively
- actual operation and observation is too disruptive
2 Types of simulation model
- Deterministic - all data known with certainty
- Probabilistic - Some data described by probability distributions
Shortcomings of simulation (5)
- Not precise
- Expensive, and timely to develop
- Not available for all situations. Without random component, all experiments would produce same answer
- Evaluates, does not generate solution techniques
- Changing too many parameters at same time
5 steps of Monte Carlo simulation
For RANDOM NUMBERs.
1. Setting up probability distribution for variables
2. Building a cumulative probability distribution for a variable
3. Establish an interval of random numbers for a variable
4. Generate random numbers
5. Simulate a series of trials
Midsquare technique for random numbers
- Start with 4 digit number
- Square it to get 8 digit (add 0 to front if 7)
take middle 4 (digit 3, 4, 5, and 6) and /1000 for random number - The middle 4 digits is the next base number
Congruential random number generator
Random number is a function of seed mod m.
eg. seed Z0 =1, f(Z)=aZ0 mod m
a = 6, m=13
Z1 = (1*6)/13 = 0 remain 6.
6/m-1= 6/12=0.5
0.5 is the random number.
Why use inventory management models
Help managers face problems of maintaining sufficient inventories to meet demand as well as incurring lowest inventory holding costs
Inventory management costs to consider
- Ordering costs
- Holding costs
- (Backorder cost)
What is Simulation?
The process of building a mathematical or logical model of a system or a decision problem and experimenting with the model to obtain insights into the system’s behaviour or to assist in solving the decision problem.
Steps in Simulation Process (7)
- Define the problem/system, identifying the basis entities.
- Formulate the model (i.e., process and flows).
- Identify/collect data to test the model.
- Test the model: compare the behaviour of the model with actual problem.
- Run the simulation.
- Analyse the results of the simulation.
- Validate the simulation.
Formula to calculate x for a Uniform Probability Distribution
Simulation - Random Numbers
x = a + R(b - a)
Formula to calculate x for a Normal Probability Distribution
x = mu +/- SD(z)
From the fomula: z = (x - mu)/SD
Alternative Approaches to Inventory Management
- Deterministic Inventory Models - rate of demand is constant or near constant.
- Probabilistic Inventory Models - demand for items fluctuates.
- Material Requirement Planning (MRP) - where demand is directly dependent on demand of final products in inventory system.
- Just-in-time (JIT) - eliminating waste, including unnecessary inventory.
What is the Economic Order Quantity (EOQ)?
The number of units that a company should add to inventory with each order to minimise the total costs of inventory.
Model for calculating the appropriate reorder point, and optimal reorder quantity.
Constraints to EOQ model
In terms of Demand and Quantity.
- Demand (D) for an item is constant per time period.
- Quantity (Q) ordered arrives at one point in time - inventory level then decreases until the next re-order point.
What is the equation for adjusted reorder point for Safety Stock?
rss = (d x m) + SS
EOQ model Assumptions
(Economic Order Quantity)
- Demand is deterministic and occurs at a constant rate.
- The order quantity is the same for every order, and it all turns up at once.
- The cost per order is constant, does not depend on the quantity ordered.
- The purchase cost per unit is constant and does not depend on quantity ordered.
- The inventory holding cost per unit per time period is constant.
- Shortages, such as stockouts or backorders are not permitted.
- The lead time for an order is constant.
- The inventory position is reviewed continuously. As a result, an order is placed as soon as the inventory positioon reaches the reorder point.
What is the Economic Production Lot Size Model?
Goods arrive at a constant rate (p) over a full period of time.
Disadvantages of Quantity Discounts for the EOQ Model
- higher ordering costs
- older stock - becomes unhealthy to eat, but also the volatility of prices of component parts of technology.
- more money upfront
What are the 3 levels of Management Science?
- Fundamental Level - Probability, Optimisation and Dynamical systems.
- Modelling Level - building models, analysing them mathematically, gathering and interpreting data.
- Application Level - make a practical impact and be a driver for change in the real world.
