Question

A small car dealer, who is eager to estimate his inventory cost, can hold up to 4 cars in the showroom. The periodic demand for the cars is following a Poisson distribution with mean 2, except for the maximum inventory level. In the case of maximum inventory level, the dealer makes a special discount by offering a price much below the market which results in depletion of its entire inventory. Once all the cars are sold, the dealer immediately orders 3 or 4 cars with equally likely probabilities. Apart from maximum and minimum inventory levels, the dealer can sell all his cars and also seek to refill its inventory. The Inventory cost is given as; $500 per car per period.

(a) Construct the transition probability matrix, by properly defining the states and the state space.

(b) Find the average inventory holding cost per period.

Answer 1

Let the random variable : number of cars in the inventory at time t

Then the state space of

The transition matrix is as follows:

where is the probability of transitioning to state X=j from state X=i

Once all the cars are sold, the dealer immediately orders 3 or 4 cars with equally likely probabilities

Now let's the consider the transition probabilities :

when there is no demand, the number of vehicle remain unsold
The probability of demand follows a poisson distribution,

so

now consider the transition probabilities :

= Probability of demand is exactly equal to i - j

Note:

So that,

But note that the transition from state i = 4 and j = 0 can happen when the demand exceeds 4,
so thate

Now consider the transition probabilities :

Note that

• when i = 0, we already calculated
  
• the transition from any state  
  will happen when the demand is exactly j, so that
  This gives us:
  
  
  
• the transition from any state to a state  , happens when the
  demand is 4 or more than 4, so that
  
  This gives us:
  

So finally this gives us the transition matrix as follows:

(b) The average inventory holding cost per period =

, where is the stationary distribution of the given Stochastic process chain

The stationary distribution \pi can be found by the eigen decomposition of the transition matrix

you can do this in R software using the following command:

> P <- matrix(c(0, 0, 0, 0.5, 0.5, 0.2706, 0.1353, 0.2706, 0.18044, 0.1428, 0.2706, 0.2706, 0.1353, 0.18044, 0.1428, 0.18044, 0.2706, 0.2706, 0.1353, 0.1428, 0.1428, 0.18044, 0.2706, 0.2706, 0.1353), nrow=5, ncol=5, byrow=T) > P [,1] [,2] [,3] [,4] [,5] [1,] 0.00000 0.00000 0.0000 0.50000 0.5000 [2,] 0.27060 0.13530 0.2706 0.18044 0.1428 [3,] 0.27060 0.27060 0.1353 0.18044 0.1428 [4,] 0.18044 0.27060 0.2706 0.13530 0.1428 [5,] 0.14280 0.18044 0.2706 0.27060 0.1353 > stnry.distr = Re(eigen(t(P))$vectors[, 1]) > stnry.distr <- stnry.distr/sum(stnry.distr) > stnry.distr [1] 0.1751886 0.1804367 0.1966319 0.2438510 0.2038919

Hence  

This gives the average inventory holding cost per period =

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