Wednesday, November 01, 2006

Deterministic and Probabilistic Models

To understand it better, let us visualize deterministic and probabilistic situations.

A deterministic situation is one in which the system parameters can be determined exactly. This is also called a situation of certainty because it is understood that whatever are determined, things are certain to happen the same way. It also means that the knowledge about the system under consideration is complete then only the parameters can be determined with certainty. At the same time you also know that in reality such system rarely exists. There is always some uncertainty associated.

Probabilistic situation is also called a situation of uncertainty. Though this exists everywhere, the uncertainty always makes us uncomfortable. So people keep trying to minimize uncertainty. Automation, mechanization, computerization etc. are all steps towards reducing the uncertainty. We want to reach to a situation of certainty.

Deterministic optimization models assume the situation to be deterministic and accordingly provide the mathematical model to optimize on system parameters. Since it considers the system to be deterministic, it automatically means that one has complete knowledge about the system. Relate it with your experience of describing various situations. You might have noticed that as you move towards certainty and clarity you are able to explain the situation with lesser words. Similarly, in mathematical models too you will find that volume of data in deterministic models appears to be lesser compared to probabilistic models. We now try to understand this using few examples.

Take an example of inventory control. Here there are few items that are consumed/ used and so they are replenished too either by purchasing or by manufacturing. Give a thought on what do you want to achieve by doing inventory control. You may want that whenever an item is needed that should be available in required quantity so that there is no shortage. You can achieve it in an unintelligent way by keeping a huge inventory. An intelligent way will be to achieve it by keeping minimum inventory. And hence, this situation requires optimization. You do this by making decisions about how much to order and when to order for different items. These decisions are mainly influenced by system parameters like the demand/ consumption pattern of different items, the time taken by supplier in supplying these items, quantity or off-season discount if any etc. Let us take only two parameters -- demand and time taken by supplier to supply, and assume that rest of the parameters can be ignored.

If the demand is deterministic, it means that it is well known and there is no possibility of any variation in that. If you know that demand will be 50 units, 70 units and 30 units in 1st, 2nd and 3rd months respectively it has to be that only. But in a probabilistic situation you only know various possibilities and their associated probabilities. May be that in the first month the probability of demand being 50 units is 0.7 and that of it being 40 units is 0.3. The demand will be following some probability distribution. And you can see that the visible volume of data will be higher in case of probabilistic situation.

You have different mathematical models to suit various situations. Linear Programming is a deterministic model because here the data used for cost/ profit/ usage/ availability etc. are taken as certain. In reality these may not be certain but still these models are very useful in decision making because

1. It provides an analytical base to the decision making
2. The sensitivity of performance variables to system parameters is low near optimum.
3. Assuming a situation to be deterministic makes the mathematical model simple and easy to handle.

But if the uncertainty level is high and assuming the situation to be deterministic will make the model invalid then it is better to use probabilistic models. Popular queuing models are probabilistic models as it is the uncertainty related to arrival and service that form a queue.

4 Comments:

Blogger Rajib Roy said...

Girijesh,
Really liked your blog. I happen to work in a company that uses probablistic models to solve inventory problems. You will be amazed how few customers get the "managing variability through probabilistic models" concept still. However, I perceive that it is changing - slowly but surely - mostly led by High Tech and CPG industries. Check out www.optiant.com some time...

Rajib

6:17 AM  
Anonymous Girijesh said...

Dear Rajib,

Thanks for your comment. I agree with you. The optimization techniques have great power to better the way of decision making. But very few of the decision makers are aware of that. I have visited your website and feel that organization like these can bridge the gap between operations research in theory and operations research in practice. Surprisingly this can be achieved by taking simpler steps than what one has been taking.

Girijesh

1:40 AM  
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