Statistical Inventory control models I by tym76564

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									Statistical Inventory
  control models I


      (Q, r) model
Learning objective
   After this class the students should be able to:

    •   Apply optimization technique to inventory model
    • calculate the appropriate order quantity in the
        face of uncertain demand.
    •   Derive the (Q, r) model from the integration of the
        EOQ model and the EPL model.
    •   analyze the implication of the (Q, r) model
Time management
   The expected time to deliver this module
    is 50 minutes. 30 minutes are reserved
    for team practices and exercises and 20
    minutes for lecture.
The Base Stock Model
   Consider the situation facing an appliance store that sells a
    particular model of refrigerator. Because space is limited and
    because the manufacturer makes frequent deliveries of other
    appliances, the store finds it practical to order replacement
    refrigerators each time one is sold. In fact, they have a system
    that places purchase orders automatically whenever a sale is
    made. But, because the manufacturer is slow to fill
    replenishment orders, the store must carry some stock in order
    to meet customer demands promptly. Under these conditions,
    the key question is how much stock to carry.
Assumptions
1.   Demands occur one at a time;

2.   Any demand not filled from stock is backordered;

3.   Replenishment lead times are fixed and known;

4.   There is no setup cost associated with placing an order; and

5.   There is no constraint on the number of orders that can be
     placed per year.

    Last two assumptions imply that there is no incentive to
     replenish stock in anything other than one-at-a-time fashion.
 Notation
l  Replenishm ent lead - time (in years)
X  Demand during replenishm ent (in units), a random variable
G  x   P X  x  cumulative distributi on function of demand
                 during replenishm ent lead - time;
  E  X  Demand (in units) during lead - time l ;
h  Cost (in dollar per unit per year) to carry one unit of inventory for one year;
b  Cost (in dollar per unit per year) to carry one unit backorder for one year
r  reorder point (in units), which represents the inventory level that trigger
     a replenshment order; this is the decision v ariable;
R  r  1, base stock level (in units)
s  r   , base stock level (in units)
The question
   We place an order when there are r units in stock and
    expect to incur demand for θ units while we wait for the
    replenishment order to arrive. Hence, r - θ is the amount
    of inventory we expect to have on hand when the order
    arrives. If s = r - θ > 0, then we call this the safety stock
    for this system, since it represents inventory that protects
    it against stockouts due to fluctuations in either demand
    or deliveries. Since finding r - θ is equivalent to finding r
    (because θ is a constant), we can view the problem either
    as finding the optimal base stock level (R = r -1), reorder
    point (r), or safety stock level (s = r - θ).
Solution
   Since an order is placed every time a demand
    occurs, the relationship “on-hand + purchase
    orders - backorders = R” holds at all times.

   Because lead times are constant, we know that all of
    the other R-1 items in inventory and on order will be
    available to fill new demand before the order under
    consideration arrives.

   Therefore, the only way the order can arrive after the
    demand for it has occurred is if demand during the
    replenishment lead time is greater than or equal to R
    (i.e., X ≥ R).
Solution
   Hence, the probability that the order arrives before
    its demand (i.e., does not result in a backorder) is
    given by P(X < R). If demand has a continuous
    distribution then

             P(X < R) = P(X ≤ R-1) = G(R).

   However, if demand has a discrete distribution (i.e.,
    X can take on only integer values and has a
    probability mass function instead of a density
    function), then

           P(X < R) = P(X ≤ R-1) = G(R-1)=G(R).
Solution
   Since all orders are alike with regard to this
    calculation, the fraction of demands that are
    filled from stock is equal to the probability that
    an order arrives before the demand for it has
    occurred, or

                     G ( R) if demand is continous
        P X  R   
                     G r  if demand is discrete

    Hence, G(R) or G(r) represents the fraction of
     demands that will be filled from stock (i.e., the fill
     rate).
The (Q, r) model
   Consider the situation where inventory is
    monitored continuously and demands occur
    randomly, possibly in batches. When the
    inventory level reaches (or goes below) r, an
    order of size Q is placed. After a lead-time of ℓ,
    during which a stockout might occur, the order
    is received. The problem is to determine
    appropriate values of Q and r. The model we
    use to address this problem is known as the
    (Q, r) model
Example
   The manager of a maintenance department must stock spare parts to
    facilitate equipment repairs.

   Demand for parts is a function of machine breakdowns, therefore
    random.

   Unlike the base stock model, the costs incurred in placing a purchase
    order (for parts obtained from an outside supplier) or the costs
    associated with setting up the production facility (for parts produced
    internally) are significant enough to make one-at-a-time replenishment
    impractical.

   Thus, the maintenance manager must determine not only how much
    stock to carry (as in the base stock model), but also how many to
    produce/order at a time (as in the EOQ and newsboy models).
Assumptions
   From a modeling perspective, the (Q, r) model is identical
    to the base stock model, except that we will assume that
    either

    •   There is a fixed cost associated with a replenishment order.
    •   There is a constraint on the number of replenishment orders
        per year.

   Therefore, replenishment quantities greater than one may
    make sense.
Basic mechanics of the (Q, r) model
Demands occur randomly, possibly in batches. When the inventory level
reaches (or goes below) the reorder point r, a replenishment order for
quantity Q is placed. After a (constant) lead time of ℓ, during which a
stockout might occur, the order is received. The problem is to determine
appropriate values of Q and r.
Total cost
In some sense, the (Q, r) model represents the integration of the EOQ
model and the base stock model two models. Then, the total cost is




                    Q        
   Y Q, r   A  h  r     b n(r )
              D                    D
              Q     2            Q
 Where n(r) is the expected number of backorders that
 will be placed during a cycle
Optimal replenishment quantity
   The optimal
    replenishment
    quantity Q*, and       2 D( A  bn(r )
                        Q
    reorder point r*,             h
    can be found by
    simultaneously
                                     hQ
    solving the         G ( R)  1 
    following                        bD
    equations:
Reflections
Each team is invited to analyze the
   following insights, based on the
   statistical model (10) minutes):

1.   “Cycle stock increase as replenishment
     frequency decrease”
2.   “Safety stock provide a buffer against
     stockout”
Reflections
   Suppose you are stocking parts
    purchased from vendors in a warehouse.
    How could you use a (Q, r) model to
    determine whether a vendor of a part
    with a higher price but a shorter lead
    time is offering a good deal? What other
    factors should you consider in deciding
    to change vendors. (10 minutes)
Reflections
   What is the key difference between the
    EOQ model and the (Q, r) model?
    Between the base stock model and the
    (Q, r) model?
Reference
   Factory Physics. Hopp & Spearmen,
    Irwin, 1996. Chapter 2, p.72-100

								
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