Inventory Control- Insensitivity of Total Cost near Optimum

You must have studied the famous EOQ (Economic Order Quantity) formula—

EOQ = Sqrt (AD/ VR) where Sqrt means square root of, A = Ordering cost/ order, D = total demand or consumption or requirement, V = unit cost of the item and R = rate of inventory carrying cost. This is a deterministic model because all these variables are assumed to be certain.

EOQ is economic order quantity because, if this quantity is ordered in every order of the item then the total cost is minimum. If you order for lesser quantity than EOQ then you will have to order more number of times to meet the same demand D, thus increasing one component of the total cost. If you order for more quantity than EOQ, then inventory level will become higher and so the inventory carrying cost will increase. The total of these two costs i.e. ordering cost and inventory carrying cost is minimum if EOQ is used as order quantity.

You know that if values of A, D, V and R are known then you substitute the values in the formula to get EOQ. But have you thought how the value of these variables is obtained? The total demand or requirement or consumption (D) may be obtained by using suitable forecasting technique. Ordering cost/ order (A) may be obtained by using previous data. Similarly you may analyse the data of various components of inventory carrying cost to get the value of R. But all these cannot be considered as deterministic. No forecasting technique can forecast with certainty what the demand would be. Then, how can this formula be used in a real life scenario? And what sense it will make to have a data on EOQ where all the variables used in computation are likely to change? Estimating the value of R, A and D involves cost and one cannot keep doing it before every order.

But there is a general characteristic of optimum solution in every field. The performance becomes insensitive to changes in effort near the optimum level. If you see the graph between Total Cost and Order Quantity, you find that the curve is flat near EOQ. It means that if you change the order quantity but keep it near the EOQ value then the minimum or optimum total cost will remain almost same. A relatively larger change in order quantity will change the total cost very little if the order quantity is close to EOQ. But if you operate away from EOQ, then a small change in order quantity can bring a big change in total cost. That is why if a good student is performing much below his optimum level then a small improvement in effort can bring big improvement in performance. But once he reaches closer to his optimum level he needs bigger effort to bring improvement in performance.

In inventory model, insensitivity plays a very important role. It permits you to make errors in estimation of A, D, V and R as long as you use EOQ and try to be near the optimum. These errors will not be affecting the total cost much so you can afford to use simpler methods in estimation. But if you operate away from the optimum level which you will if scientific management techniques are not used, then the total cost will be very sensitive to order quantity and error in estimation can adversely affect the total cost drastically.

Hence, in spite of all the possibilities of uncertainty, errors etc. this deterministic model is of immense use in keeping the total cost near the minimum possible level. Uncertainty and error in estimation factors are taken care by building system of automatic response to these in the method of inventory control itself. This will be discussed in some other article.

Sometimes we hear an Inventory Manager saying ‘whenever the current stock goes below some figure say 100, we order for the item a quantity equivalent to last 3 months consumption and this method works fine for us’. There is no harm in agreeing to him on similar statements. In fact it is not necessary to do any complex mathematical modeling and computation to reach to an optimum decision. Concept of optimization techniques can be built in applications in much simpler ways. We will discuss this too in some other article.

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.