Lecture No 18

Held on: Monday, September 25, 2000

Done by sachin bhalla entry no. 97270

STOCHASTIC INVENTORY PROBLEM

Cu : cost of unsatisfied demand / unit

Co : cost of positive inventory / unit

F(Q*) = Cu/(Cu + Co)

            = 50/65 = 0.77

D ~ N (11.73 , 4.74)

P(D <= Q*) = 0.77

X ~ N(µ , s ²)

X - µ ~ N(0 , 1)

s

P (N(0 , 1) <= Q* - 11.73) = 0.77

                         (4.74)^1/2         

 

 

 

 

FIG 1

Q* - 11.73 = Z* = 0.74

(4.74)^1/2

 

Q* = s Z* + µ

= 15.24 ~ 15 newspapers        

E(cost) = Co _0ò ^Q (Q-X) f(X) dX + _{Qò ¥} (X-Q)f(X) dX

Cost price = c

Inventory holding cost = h

Sale price = s

G(Q) = cQ + h_0ò ^Q (Q-X) f(X) dX – s_0ò ^Q X f(X) dX – sQ_{Qò ¥} f(X)dX +(s-c)_{0ò ¥} Xf(X)dX

= (c+h)_0ò ^Q(Q-X)f(X)dX + (s-c)_Qò ^¥ (X-Q)f(X)dX

c+h = Co , s-c = Cu

if starting inventory is u >= 0

then order Q* - u if u < Q*

            do not order if u >= Q*

he orders upto level Q

find optimal Q                           it turns out to be Q*

 

 

FIG 2

(s,S) policy

if inv. Is less then s then order S units

G(Q) = Ec(Q,D)

C(Q,D) : actual cost if demand is D and order amount is Q

G(s) = k + G(S)

Question: Bicycle production has been discontinued. Stores are informed that no reorders are possible

Cost of each bicycle = Rs 2000

Cost of maintainence & inv. = Rs 100

Salvage cost = Rs 1000

Sale price = Rs 4500

ED = 10000

F(X) =e^-X/10000

            10000

 

Answer : F(Q*) = 2500/(2500+1100) =0.6944

₀ ò ^Q* e^-X/10000dX = 0.6944

10000

Q* = 11856

If set up cost k = Rs 80,000

G(s) = 2500_{sò ¥} (X-s) f(X)dX +1100_0ò ^s(s-X) f(X)dX

=k +G(S)

= 80,000 + 2500_11856ò ^S(X-11856)f(X)dX + 1100_Qò ¹¹⁸⁵⁶(11856 – X)f(X)dX

s = 10,674