Inventory Models-deterministic Models

Probabilistic or stochastic models: The situation is known as Models with unknown demand or models with probabilistic demand.

·         This means that demand can be known with certain probability. When the probability of demand ‘r’ is expected, then we cannot minimize the actual cost.

·         But the optimal quantity of inventory is determined on the basis of minimizing the total expectedcost represented by (TEC) instead of minimizing the actual cost.

·         To face these uncertainties in consumption rate and lead time, an extra stock is maintained to meet the demand, in case any shortage is there. The extra stock is termed as BUFFER STOCK OR SAFETY STOCK.

        Single period model with uniform demand (No set up cost model) In this model the following assumptions are made:

(a)Reorder time is fixed and known say ‘t’ units of time. Therefore the set up cost C3 is not included in the total cost.

(b)Demand is uniformly distributed over period. Here the term period refers for the time of one cycle.

(c)Production is instantaneous, i.e. lead-time is zero.

(d)Shortages are allowed and they are backlogged. The costs included in this model are C1 carrying cost per unit of quantity per unit of time and C2 the shortage cost per unit of quantity per unit of time.

(e)Units are discrete and p(r) is the probability of requiring ‘r’ units per period. If ‘S’ is the level of inventory in the beginning of each period, and we have to find the optimum value of ‘S’. Hence the decision variable is S.

In this problem two situations will arise:

(a) Demand r ≤ S, (b) demand r > S. The two situations are illustrated by means of graph in Inventory in one cycle = ½ (S S – r) t = ½ (2S – r) t = (S – r/2) t. units. Hence inventory Carrying cost = C1 × (S – r/2) × t, this is true when r ≤ S. But the demand is equal to ‘r’ is with a probability of p (r). Hence the expected carrying cost = C1 × (S – r/2) × t × p (r).

Example: A contractor of second hand motor trucks uses to maintain a stock of trucks every month. The demand of the trucks occurs at a constant rate but not in constant size. The probability distribution of the demand is as shown below:

The holding cost of an old truck in stock for one month is /Rs.100/- and penalty for a truck if not delivered to the demand, is Rs. 1000/-. Determine the optimal size of the stock for the contractor.

Solution