Lecture No17

Held on: Wednesday, September 20, 2000

Prepared by: Anil Kumar (97231)

 

                                  Stochastic Inventory Theory

  Newsboy problem

Assume that

Demand N(11.73,4.74) i.e., mean is 11.73and variance is 4.74

    

 

 

 

 

         f(x)

 

 

 

 

 

 

 

 

 

 

                                                 m=11.73

 

The probability distribution function for a normal distributed random variable is given by

f(x) =    1     exp(-(x-m)²/2s² )

                   (2p)^1/2s

Probability of demand between  x₁ and x₂ is given by

          P(x₁<D<x₂) = _x1ò^x2 f(x)dx

 

Consider an example

          Each newspaper copy is purchased for 25p

          Each newspaper copy sold for 75p

          Any newspaper copy is returned for 10p

Find out the number of newspapers that the newsboy should buy to maximize his expected  profits.

 

 This type of problem can be related to the following areas

            1.            product that perishes quickly

            2. short lived style goods

3.     newsboy problem

 

let      C_o = Cost per unit of positive inventory remaining at the end of period.(overage cost)

Cu=Cost per unit in satisfied demand.(underage cost)

Co=15

Cu=50

 

 

D is a continuous random variable with pdf f(x) and c.d.f. F(x)

random no.  maps outcome to a real number

X is a continuous  random variable then

                    P (xÎ(a,b))=_aò^b f(x)dx

If there exists a pdf f(x) such that F(x) = _-¥ò^x f(x)dx= P(X<x)

 

 

To find optimum Q

1.     Develop the cost incurred as a function as a function of D&Q.

2.     Determine the expected value of this cost.

3.     Determine the expected value of this Q that minimizes the cost function

 

 

Step1:

If newsboy purchases Q_units  at the beginning of the period and demand for the newspapers turn out to be D.

          Then the loss is given by

                   G(Q,D) = (Q-D)⁺C₀ + (D-Q)⁺C_u

 

          Where X+ = X if X³0

                          = 0  if X<0

 

Our objective is to minimize expectation of G(Q,D) represented as EG(Q,D)

Step2:

Expected cost function is given by

          C(Q)= Eg(x) = ₀ò^¥ g(x)f(x)dx

 

Therefore EG(Q,D) = C_{o 0}ò^Q Q-x)f(x)dx + C_{u Q}ò^¥ (x-Q)f(x)dx

 

 

Step3:

        At optimum Q d(Q) =0

                               dQ

Differentiating w.r.t. Q and equating to zero

          d(EG(Q,D)) = 0

                  dQ

At optimum value of Q=Q* the c.d.f. is

          F(Q*) =       Cu     .

                         Cu + Co        (critical ratio )

 

 

 

 

 

 

 

 

 

 

 

            C_u     .

C_u+C_o                                                                                                                                                                                                                                                                                                                                                                                                                                 F(Q)                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     Q*

                                                                             Qà