Outline Multi-Period Models Lot size-Reorder Point ( Q , R ) Systems

Outline

• Multi-Period Models – Lot size-Reorder Point (Q, R) Systems

• Optimization without service– Procedure– Example

LESSON 18: INVENTORY MODELS (STOCHASTIC)Q,R SYSTEMS

OPTIMIZATION WITHOUT SERVICE

Goal: Given find (Q,R) to minimize total cost

Step 1: Take a trial value of Q = EOQ

Step 2: Find a trial value of R = where and are respectively mean and standard deviation of the lead-time demand and is the normal distribution variate corresponding to the area on the right, 1-F(z) = see Table A-4, pp. 835-841

Step 3: Find the expected number of stock-outs per cycle, where is the standardized loss function available from Table A-4, pp. 835-841

z

z

pQh /

)(zLn )(zL

pKh ,,,,

Procedure to find the Optimal (Q,R) Policy Without Any Service Constraint

Procedure to find the Optimal (Q,R) Policy Without Any Service Constraint

Step 4: Find the modified

Step 5: Find the modified value of R = where is the recomputed value of the normal distribution variate corresponding to the area on the right, 1-F(z) = see Table A-4, pp. 835-841

Step 6: If any of modified Q and R is different from the previous value, go to Step 3. Else if none of Q and R is modified significantly, stop.

Knph

Q 2

z z

pQh /

Example - Optimal (Q,R) Policy

Annual demand for number 2 pencils at the campus store is normally distributed with mean 2,000 and standard deviation 300. The store purchases the pencils for 10 cents and sells them for 35 cents each. There is a two-month lead time from the initiation to the receipt of an order. The store accountant estimates that the cost in employee time for performing the necessary paper work to initiate and receive an order is $20, and recommends a 25 percent annual interest rate for determining holding cost. The cost of a stock-out is the cost of lost profit plus an additional 20 cents per pencil, which represents the cost of loss of goodwill. Find an optimal (Q,R) policy


y

y

p

Ich

K

demand, time-lead of deviation Standard

demand, time-lead Mean

time, Lead

demand, annual of deviation Standard

demand, annual Mean

cost,Penalty

cost, Holding

cost, ordering Fixed


I t e r a t i o n 1 S t e p 1 :

h

KQ

2EOQ

S t e p 2 :

p

QhzF )(1

z ( T a b l e A - 4 )

zR


S t e p 3 : )( zL ( T a b l e A - 4 ) )( zLn

S t e p 4 :

Knph

Q2

Q u e s t i o n : W h a t a r e t h e s t o p p i n g c r i t e r i a ?

S t e p 5 : p

QhzF )(1

z ( T a b l e A - 4 )

zR

Example - Optimal (Q,R) Policy Iteration 2S t e p 3 : )( zL ( T a b l e A - 4 )

)( zLn S t e p 4 :

Knph

Q2

Q u e s t i o n : D o t h e a n s w e r s c o n v e r g e ?

S t e p 5 : p

QhzF )(1

z ( T a b l e A - 4 )

zR

Fixed cost (K) Note: K, h, and p Holding cost (h) are input dataPenalty cost (p)Mean annual demand () inputLead time (in years input dataLead time demand parameters:

Mean, <--- computedStandard deviation, input data

Iteration 1Iteration 2Step 1Q=Step 2Area on the right=1-F(z)

z=R=

Step 3L(z)=n=

Step 4Modified Q=Step 5Area on the right=1-F(z)

z=Modified R =

z

z

EOQ

pQh /

pQh /

Table A1/A4

Table A1/A4

Table A4

)(zL hKnp /2

READING AND EXERCISES

Lesson 18

Reading:

Section 5.4, pp. 262-264 (4th Ed.), pp. 253-255 (5th Ed.)

Exercise:

13a, p. 271 (4th Ed.), p. 261

Outline Multi-Period Models Lot size-Reorder Point ( Q , R ) Systems

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