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Deterministic Inventory Management

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Deterministic Inventory Management Powered By Docstoc
					 Order Quantities when
Demand is Approximately
         Level
          Chapter 5
   Inventory Management
   Dr. Ron Tibben-Lembke
           Inventory Costs

Costs associated with inventory:
   Cost of the products
   Cost of ordering
   Cost of hanging onto it
   Cost of having too much / disposal
   Cost of not having enough (shortage)
                Shrinkage Costs

   How much is stolen?
       2% for discount, dept. stores, hardware,
        convenience, sporting goods
       3% for toys & hobbies
       1.5% for all else
   Where does the missing stuff go?
       Employees: 44.5%
       Shoplifters: 32.7%
       Administrative / paperwork error: 17.5%
       Vendor fraud: 5.1%
    Inventory Holding Costs

Category               % of Value
 Housing (building) cost        6%
 Material handling              3%
 Labor cost                     3%
 Opportunity/investment         11%
 Pilferage/scrap/obsolescence   3%
Total Holding Cost              26%
                ABC Analysis
   Divides on-hand inventory into 3 classes
       A class, B class, C class
   Basis is usually annual $ volume
       $ volume = Annual demand x Unit cost
   Policies based on ABC analysis
     Develop class A suppliers more
     Give tighter physical control of A items

     Forecast A items more carefully
                Classifying Items
                     as ABC
% Annual $ Usage
  100
   80
   60
   40
            A
   20              B               C
    0
        0           50          100      150
                  % of Inventory Items
  ABC Classification Solution

Stock #   Vol.     Cost    $ Vol.     %   ABC
 206      26,000   $ 36 $936,000
 105         200    600 120,000
 019       2,000     55 110,000
 144      20,000      4   80,000
 207       7,000     10   70,000
 Total                    1,316,000
  ABC Classification Solution

Stock #   Vol.     Cost    $ Vol.    %      ABC
 206      26,000   $ 36 $936,000     71.1   A
 105         200    600 120,000       9.1   A
 019       2,000     55 110,000       8.4   B
 144      20,000      4   80,000      6.1   B
 207       7,000     10   70,000      5.3   C
 Total                    1,316,000 100.0
     Economic Order Quantity

Assumptions
   Demand rate is known and constant
   No order lead time
   Shortages are not allowed
   Costs:
     A - setup cost per order
     v - unit cost

     r - holding cost per unit time
                    EOQ
   Inventory
   Level


Q*             Decrease Due to
Optimal        Constant Demand
Order
Quantity




                                 Time
               EOQ
   Inventory
   Level


Q*             Instantaneous
Optimal        Receipt of Optimal
Order          Order Quantity
Quantity




                                    Time
                     EOQ

      Inventory
      Level
 Q*



Reorder
Point
(ROP)

                           Time

             Lead Time
                     EOQ

      Inventory
      Level
 Q*
                           Average
                           Inventory Q/2
Reorder
Point
(ROP)

                              Time

             Lead Time
               Total Costs

   Average Inventory = Q/2
   Annual Holding costs = rv * Q/2
   # Orders per year = D / Q
   Annual Ordering Costs = A * D/Q
   Annual Total Costs = Holding + Ordering
                    Q      D
       TC (Q)  vr *  A *
                    2      Q
         How Much to Order?

Annual Cost




                          Holding Cost
                          = H * Q/2



                       Order Quantity
         How Much to Order?

Annual Cost

       Ordering Cost
       = A * D/Q



                          Holding Cost
                          = H * Q/2



                       Order Quantity
         How Much to Order?
                     Total Cost
Annual Cost          = Holding + Ordering




                         Order Quantity
         How Much to Order?
                          Total Cost
Annual Cost               = Holding + Ordering




              Optimal Q       Order Quantity
                Optimal Quantity

                          Q      D
Total Costs =         vr *  A *
                          2      Q
                Optimal Quantity

                          Q      D
Total Costs =         vr *  A *
                          2      Q
Take derivative       vr     D
with respect to Q =       A* 2
                      2      Q
                Optimal Quantity

                          Q      D
Total Costs =         vr *  A *
                          2      Q
Take derivative       vr     D        Set equal
with respect to Q =       A* 2  0   to zero
                      2      Q
                Optimal Quantity

                          Q      D
Total Costs =         vr *  A *
                          2      Q
Take derivative       vr     D        Set equal
with respect to Q =       A* 2  0   to zero
                      2      Q
Solve for Q:

 vr DA
    2
 2 Q
                Optimal Quantity

                          Q      D
Total Costs =         vr *  A *
                          2      Q
Take derivative       vr     D        Set equal
with respect to Q =       A* 2  0   to zero
                      2      Q
Solve for Q:

 vr DA               2 AS
    2           Q 
                  2
 2 Q                  vr
                Optimal Quantity

                          Q      D
Total Costs =         vr *  A *
                          2      Q
Take derivative       vr     D        Set equal
with respect to Q =       A* 2  0   to zero
                      2      Q
Solve for Q:

 vr DA               2 AS             2 AS
    2           Q 
                  2
                                Q
 2 Q                  vr               vr
                   Sensitivity

   Suppose we do not order optimal EOQ, but
    order Q instead, and Q is p percent larger
   Q = (1+p) * EOQ
   Percentage Cost Penalty given by:
                    p2 
           PCP  50     
                   1 p 
                        
   EOQ = 100, Q = 150, so p = 0.5
    50*(0.25/1.5) = 8.33 a 8.33% cost increase
              Figure 5.3 Sensitivity
             Percentage Cost Penalty using Q different from the EOQ

                                      30


                                      25


                                      20


                                      15
PCP




                                      10


                                       5


                                       0
      -0.6      -0.4        -0.2           0        0.2         0.4   0.6
                                      -5
                                           p
              A Question:

   If the EOQ is based on so many
    horrible assumptions that are never
    really true, why is it the most
    commonly used ordering policy?
            Benefits of EOQ

   Profit function is very shallow
   Even if conditions don’t hold
    perfectly, profits are close to optimal
   Estimated parameters will not throw
    you off very far
            Tabular Aid 5.1

   For A = $3.20 and r = 0.24%
   Calculate Dv =total $ usage (or sales)
   Find where Dv fits in the table
   Use that number of months of supply
   D = 200, v = $16, Dv=$3,200
   From table, buy 1 month’s worth
   Q = D/12 = 200/12 = 16.7 = 17
      How do you get a table?

   Decide which T values you want to
    consider: 1 month, etc.
   Use same v and r values for whole table
   For each neighboring set of T’s, put them
    into

                  288 A
             Dv 
                  T1T2 r
      How do you get a table?

   For example, A = $3.20, r = 0.24
   To find the breakpoint between 0.25 and 0.5
   Dv = 288 * 3.2 / (0.25 * 0.5 * 0.24)
   = 921.6 / 0.03 = 30,720
   So if Dv is less than this, use 0.25, more
    than that, use 0.5
   Find 0.5 and 0.75 breakpoint:
   Dv = 288 * 3.2/(0.5 * 0.75 * 0.24) = 10,2240
        Why care about a table?
   Some simple calculations to get set up
   No thinking to figure out lot sizes
   Every product with the same ordering cost
    and holding cost rate can use it
   Real benefit - simplified ordering
       Every product ordered every 1 or 2 weeks, or
        every 1, 2, 3, 4, 6, 12 months
       Order multiple products on same schedule:
            Get volume discounts from suppliers
            Save on shipping costs
            Savings outweigh small increase from non-EOQ orders
Uncoordinated Orders




                   Time
         Simultaneous Orders




                                                    Time
Same T = number months supply allows firm to order at
same time, saving freight and ordering expenses
Adjusted some T’s, changed order times
                Offset Orders




Same T = number months supply allows firm to control
maximum inventory level by coordinating replenishments
With different T, no consistency
          Quantity Discounts

   How does this all change if price
    changes depending on order size?
   Explicitly consider price:

               2 AD
        Q
                vr
       Discount Example

D = 10,000   A = $20     r = 20%

    Price    Quantity     EOQ
v = 5.00     Q < 500      633
    4.50     501-999      666
    3.90     Q >= 1000    716
                 Discount Pricing

Total Cost

       Price 1      Price 2           Price 3


                   X 633
                    X 666
                       X 716

             500              1,000      Order Size
                 Discount Pricing

Total Cost

       Price 1      Price 2           Price 3


                   X 633
                    X 666
                       X 716

             500              1,000      Order Size
        Discount Example

Order 666 at a time:
Hold 666/2 * 4.50 * 0.2= $299.70
Order 10,000/666 * 20 = $300.00
Mat’l 10,000*4.50 = $45,000.00 45,599.70

Order 1,000 at a time:
Hold 1,000/2 * 3.90 * 0.2=$390.00
Order 10,000/1,000 * 20 = $200.00
Mat’l 10,000*3.90 = $39,000.00 39,590.00
         Discount Model

1.Compute EOQ for each price
2.Is EOQ ‘realizeable’? (is Q in range?)
 If EOQ is too large, use lowest
 possible value. If too small, ignore.
3.Compute total cost for this quantity
4.Select quantity/price with lowest total
  cost.
          Adding Lead Time

   Use same order size Q  2 DA
                               vr

   Order before inventory depleted
   R = DL where:
     D = annual demand rate
     L = lead time in years

				
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